diff --git "a/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt" "b/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/D9AzT4oBgHgl3EQfwv5m/content/tmp_files/load_file.txt" @@ -0,0 +1,1150 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf,len=1149 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='01727v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='SI] 4 Jan 2023 Classical Solutions of the Degenerate Fifth Painlev´e Equation Peter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Clarkson School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK Email: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='Clarkson@kent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='uk January 5, 2023 Abstract In this paper classical solutions of the degenerate fifth Painlev´e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solu- tions of the degenerate fifth Painlev´e equation are known to expressible in terms of the third Painlev´e equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1 Introduction In this paper we are concerned with solutions of the equation d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(αw2 + β) z2w + γw z , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α, β and γ constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) is the special case of the fifth Painlev´e equation (PV) d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(αw2 + β) z2w + γw z + δw(w + 1) w − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α, β, γ and δ constants, when δ = 0 and is known as the degenerate fifth Painlev´e equation (deg- PV), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The six Painlev´e equations (PI–PVI), were discovered by Painlev´e, Gambier and their colleagues whilst studying second order ordinary differential equations of the form d2w dz2 = F � z, w, dw dz � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) where F is rational in dw/dz and w and analytic in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Painlev´e equations can be thought of as nonlinear analogues of the classical special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The general solutions of the Painlev´e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions and so require the introduction of a new transcendental function to describe their solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' However, it is well known that PII–PVI possess rational solutions, algebraic solutions and solutions expressed in terms of the classical special functions — Airy, Bessel, parabolic cylinder, Kummer and hypergeometric functions, respectively — for special values of the parameters, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [11, 22] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These hierarchies are usually generated from “seed solutions” using the associated B¨acklund transformations and frequently can be expressed in the form of determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These solutions of the Painlev´e equations are often called “classical solutions”, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It is well known that solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are related to solutions of the third Painlev´e equation d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + Aq2 + B x + Cq3 + D q , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) 1 with A, B, C and D constants, a result originally due to Gromak [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [22, §34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The purpose of this paper is to give a classification and description of the classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) directly, rather than indirectly through (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §2, the relationship between deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §3, classical solutions of the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) are reviewed, the rational solutions in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and the Bessel function solutions in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §4, B¨acklund transformations of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are given, which can be used to derive a hierarchy of solutions from a “seed solution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §5, classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are classified, the algebraic solutions in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and the Bessel function solutions in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In §6, two applications of classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are given to derive exact solutions of the complex sine-Gordon equation, which is equivalent to the Pohlmeyer-Lund-Regge model, and to derive explicit representations of the coefficients in the three-term recurrence relation satisfied by generalised Charlier polynomials, which are discrete orthogonal polynonials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 2 The relationship between deg-PV and PIII In the generic case when CD ̸= 0 in the third Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4), we set C = 1 and D = −1, without loss of generality (by rescaling the variables if necessary), and so consider the equation d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + Aq2 + B x + q3 − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) In the sequel, we shall refer to this equation as PIII since it is the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consider the Hamiltonian associated with PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) given by HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = q2p2 − xq2p − (2a + 2b + 1)qp + εxp + 2bxq, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with a and b parameters and ε = ±1, see [28, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then p(x) and q(x) satisfy the Hamiltonian system x dq dx = ∂HIII ∂p = 2q2p − xq2 − (2a + 2b + 1)q + εx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) x dp dx = −∂HIII ∂q = −2qp2 + 2xqp + (2a + 2b + 1)p − 2bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for p(x) gives p(x) = 1 2q � x dq dx + xq2 + (2a + 2b + 1)q − εx � , and then substituting this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) gives d2q dx2 = 1 q � dq dx �2 − 1 x dq dx + 2(a − b)q2 x + 2ε(a + b + 1) x + q3 − 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) which is PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), with parameters A = 2(a − b), B = 2ε(a + b + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for q(x) gives q(x) = 1 2p(x − p) � x dp dx − (2a + 2b + 1) + 2bx � , and then substituting this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) gives d2p dx2 = 1 2 �1 p + 1 p − x � � dp dx �2 − p x(p − x) dp dx + 2εp − 2b2 p − 4a2 − 1 2(p − x) + 1 − 4(a2 − b2) − 4εp2 2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) Then making the transformation p(x) = 2√z w(z) w(z) − 1 , x = 2√z, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) 2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) gives d2w dz2 = � 1 2w + 1 w − 1 � �dw dz �2 − 1 z dw dz + (w − 1)2(a2w2 − b2) 2z2w + εw z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2a2, β = − 1 2b2, γ = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) Hence we have the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [22, Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If q(x) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) then w(z) = xq′(x) + xq2(x) + (2a + 2b + 1)q(x) − εx xq′(x) − xq2(x) + (2a + 2b + 1)q(x) − εx, z = 1 2x2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) with ′ ≡ d/dx is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8), provided that x dq dx − xq2 + (2a + 2b + 1)q − εx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Conversely, if w(z) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8), then q(x) = 1 2√z w � z dw dz + (w − 1)(aw + b) � , x = √ 2z, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) for p(x), substituting in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) and solving for w(z) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Also solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) for q(x) and substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' An alternative method of deriving solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) involves the second-order, second-degree equa- tion satisfied associated with the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), due to Jimbo and Miwa [28] and Okamoto [46], which is often called the “σ-equation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), then σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = HIII(q, p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + qp − 1 2εx2 + (a + b)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) where q(x) and p(x) satisfy the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3), satisfies the second-order, second-degree equation (SIII) � xd2σ dx2 − dσ dx �2 + 2 ��dσ dx �2 − x2 � � xdσ dx − 2σ � − 8ε(a2 − b2)xdσ dx = 8(a2 + b2)x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) Conversely, if σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) then the solution of the Hamiltonian system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is given by q(x) = εxσ′′(x) − ε(2a + 2b + 1)σ′(x) − 2(a − b)x x2 − [σ′(x)]2 , p(x) = 1 2εσ′(x) + 1 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Jimbo and Miwa [28] and Okamoto [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) can be expressed in terms of solutions of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is a solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13), then w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + εx σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) − εx, z = 1 2x2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This immediately follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 3 3 Classical solutions of PIII and SIII 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Rational solutions of PIII and SIII Rational solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are classified in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has a rational solution if and only if ε1A + ε2B = 4n, with n ∈ Z and ε2 1 = 1, ε2 2 = 1, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Lukashevich [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Umemura [55]1 derived special polynomials associated with rational solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which we now define;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [9, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) is given by the recursion relation Sn+1Sn−1 = −x � Sn d2Sn dx2 − �dSn dx �2� − Sn dSn dx + (x + µ)S2 n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where S−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = S0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = 1, with µ an arbitrary parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) has the Wronskian representation Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = cnW (ϕ1, ϕ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' , ϕ2n−1) , cn = n � k=0 (2k + 1)n−k, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2a) where ϕm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = L(µ−2m+1) 2m−1 (−x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2b) with L(α) k (x) the Laguerre polynomial, for details see Kajiwara and Masuda [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [9, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The rational function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is given by σn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 2x d dx {ln Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)} − 1 2x2 − 2µx − 1 4, n ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) with Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) the Umemura polynomial, for the parameters a = n + 1 2, b = µ, ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Clarkson [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Special function solutions of PIII and SIII Special function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which are expressed in terms of Bessel functions and are classi- fied in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has solutions expressible in terms of the Riccati equation x dq dx = ε1xq2 + (Aε1 − 1)q + ε2x, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) if and only if ε1A + ε2B = 4n + 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) with n ∈ Z and ε2 1 = 1, ε2 2 = 1, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Further, the Riccati equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) has the solution q(x) = −ε1 d dx ln ψν(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) 1The original manuscript was written by Umemura in 1996 for the proceedings of the conference “Theory of nonlinear special functions: the Painlev´e transcendents” in Montreal, which were not published;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' for further details see [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 4 where ψν(x) satisfies xd2ψν dx2 + (1 − 2ε1ν)dψν dx + ε1ε2xψν = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) which has solution ψν(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xν {C1Jν(x) + C2Yν(x)} , if ε1 = 1, ε2 = 1, x−ν {C1Jν(x) + C2Yν(x)} , if ε1 = −1, ε2 = −1, xν {C1Iν(x) + C2Kν(x)} , if ε1 = 1, ε2 = −1, x−ν {C1Iν(x) + C2Kν(x)} , if ε1 = −1, ε2 = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) with C1, C2 arbitrary constants, and Jν(x), Yν(x), Iν(x), Kν(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Okamoto [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [11, 22, 36, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Determinantal representations of special function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) were given by Okamoto [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [19, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Suppose τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) is the determinant given by τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� x d dx �j+k ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9a) where ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9b) with c1, c2 arbitrary constants, and Jµ(z), Yµ(z), Iµ(z), Kµ(z) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is given by σn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 2x d dx {ln τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε)} + 1 2εx2 + µ2 − n2 + 2n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10a) for the parameters a = n, b = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10b) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The determinant τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) satisfies the equation x2 � τn d2τn dx2 − �dτn dx �2� + xτn dτn dx = τn+1τn−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) or equivalently � x d dx �2 ln τn = τn+1τn−1 τ 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Okamoto [46, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 4 B¨acklund transformations We note that deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has the symmetries S1 : w1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α1, β1, γ1) = w(−z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ), (α1, β1, γ1) = (α, β, −γ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) S2 : w2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α2, β2, γ2) = 1/w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ), (α2, β2, γ2) = (−β, −α, −γ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) where w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Suppose that w = w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2a2, β = − 1 2b2, γ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then wj = w(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' αj, βj, γj) given by W1 : w1 = {zw′ + (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} z2(w′)2 + 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3a) W2 : w2 = {zw′ − (w − 1)(aw − b)} {zw′ − (w − 1)(aw + b)} z2(w′)2 − 2azw(w − 1)w′ + 2cz2w(w − 1) + (w − 1)2(a2w2 − b2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3b) W3 : w3 = z2(w′)2 + 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) {zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3c) W4 : w4 = z2(w′)2 − 2bz(w − 1)w′ + 2cz2w2(w − 1) − (w − 1)2(a2w2 − b2) {zw′ − (w − 1)(aw − b)} {zw′ + (w − 1)(aw + b)} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3d) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α1 = 1 2(a + 1)2, β1 = − 1 2b2, γ1 = c, α2 = 1 2(a − 1)2, β2 = − 1 2b2, γ2 = c, α3 = 1 2a2, β3 = − 1 2(b + 1)2, γ3 = c, α4 = 1 2a2, β4 = − 1 2(b − 1)2, γ4 = c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Adler [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' also Filipuk and Van Assche [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5 Classical solutions of deg-PV To discuss classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), it is convenient to make the transformation w(z) = u(x), z = 1 2x2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which gives d2u dx2 = � 1 2u + 1 u − 1 � �du dx �2 − 1 x du dx + 4(u − 1)2(αu2 + β) x2u + 2γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) We could fix the parameter γ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), by rescaling x if necessary, but it is more convenient not to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Instead classical solutions will be classified for γ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), we have that if σ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) is a solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13), then u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) = σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) + εx σ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' a, b, ε) − εx, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with γ = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Supppose that u = u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α, β, γ) satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with parameters α = 1 2a2, β = − 1 2b2, γ = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then uj = u(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' αj, βj, γj) given by U1 : u1 = {xu′ + 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} x2(u′)2 + 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4a) U2 : u2 = {xu′ − 2(u − 1)(au − b)} {xu′ − 2(u − 1)(au + b)} x2(u′)2 − 4axu(u − 1)u′ + 4cu(u − 1)x2 + 4(u − 1)2(a2u2 − b2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4b) U3 : u3 = x2(u′)2 + 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) {xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4c) U4 : u4 = x2(u′)2 − 4bx(u − 1)u′ + 4cx2u2(u − 1) − 4(u − 1)2(a2u2 − b2) {xu′ − 2(u − 1)(au − b)} {xu′ + 2(u − 1)(au + b)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4d) 6 satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with parameters α1 = 1 2(a + 1)2, β1 = − 1 2b2, γ1 = c, α2 = 1 2(a − 1)2, β2 = − 1 2b2, γ2 = c, α3 = 1 2a2, β3 = − 1 2(b + 1)2, γ3 = c, α4 = 1 2a2, β4 = − 1 2(b − 1)2, γ4 = c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This is easily proved by applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) to B¨acklund transformations in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Algebraic solutions Algebraic solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) are equivalent to rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and so we discuss rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), which are classified in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Necessary and sufficient conditions for the existence of rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are either (α, β, γ) = � 1 2(n + 1 2), − 1 2µ2, 1 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) or (α, β, γ) = � 1 2µ2, − 1 2(n + 1 2), −1 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) where n ∈ Z and µ is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For details see Gromak, Laine and Shimomura [22, §38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We remark that the solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) are related to those satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) through the analog of the symmetry (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently we shall be concerned only with rational solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The rational solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) is given by un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = 1 − xS2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ), n ≥ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) where Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) is the Umemura polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Substituting the rational solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) and then using the reccurence relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Umemura polynomial Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) satisfies the difference equation Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = xS2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) + µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) Hence from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) there are two alternative representations of the rational solution un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) + xS2n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ), un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µSn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ + 1) Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ − 1) Sn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ)Sn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Bessel function solutions Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Necessary and sufficient conditions for the existence of Bessel function solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are either (α, β, γ) = � 1 2n2, − 1 2µ2, ε � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) or (α, β, γ) = � 1 2µ2, − 1 2n2, −ε � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) with ε = ±1, and where n ∈ Z+ and µ is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9), the parameters in PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) are given by (A, B) = � 2(a − b), 2ε(a + b + 1) � , (α, β, γ) = ( 1 2a2, − 1 2b2, ε), respectively, for parameters a, b and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The result then follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2n2, − 1 2µ2, ε � , is given by un(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 + εx2τ 2 n(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) τn+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) τn−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), n ≥ 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) where τn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� x d dx �j+k ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) and τ0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1, with ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Substituting the Bessel function solution of SIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Bessel function solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2n2, − 1 2µ2, 2ε � , is given by wn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 + εzT 2 n (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) Tn−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), n ≥ 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) where Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = det �� z d dz �j+k ψµ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) �n−1 j,k=0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) and T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1, with ϕµ(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(2√z) + c2Yµ(2√z), if ε = 1, c1Iµ(2√z) + c2Kµ(2√z), if ε = −1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='16) c1 and c2 arbitrary constants, and Jµ(x), Yµ(x), Iµ(x) and Kµ(x) Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In the next Lemma, it is shown that the first solution u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε), the “seed solution”, satisfies a first-order, second-degree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) for the parameters (α, β, γ) = � 1 2, − 1 2µ2, ε � , is u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = ϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) [xϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) − 2εµϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε)] xϕ2 µ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) − 2εµϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε)ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) + εxϕ2µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) where ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = � c1Jµ(x) + c2Yµ(x), if ε = 1, c1Iµ(x) + c2Kµ(x), if ε = −1, with c1 and c2 constants, satisfies the first-order, second-degree equation x2 �du dx �2 − 4xu(u − 1)du dx + 4εx2u(u − 1) + 4(u − 1)2(u2 − µ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Define Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) = ϕµ+1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) ϕµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) , then from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) = 1 − x εxΦ2µ − 2µΦµ + x, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) and Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) satisfies the Riccati equation xdΦµ dx = εxΦ2 µ − (2µ + 1)Φµ + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) Next we assume that u1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ, ε) satisfies a first-order, second-degree equation of the form x2 �du dx �2 + x � f2(x, µ, ε)u2 + f1(x, µ, ε)u + f0(x, µ, ε) � du dx + 4 � j=0 gj(x, µ, ε)uj = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) where {fj(x, µ, ε)}2 j=0 and {gj(x, µ, ε)}4 j=0 are to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Then substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21), using the fact that Φµ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ε) satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) and equating coefficients of powers of Φµ yields f2 = −4, f1 = 4, f0 = 0, g4 = 4, g3 = −8, g2 = 4εx2 − 4µ2 + 4, g1 = −4εx2 + 8µ2, g0 = −4µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Hence we obtain equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This demonstrates that special function solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), and hence also deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) , are different from special function solutions of PII–PVI where the “seed solution” satisfies a Riccati equation, a first- order, first-degree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 6 Applications 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 Complex sine-Gordon equation Consider the two-dimensional complex sine-Gordon equation ∇2ψ + (∇ψ)2ψ 1 − |ψ|2 + ψ � 1 − |ψ|2� = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where ∇ψ = (ψx, ψy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Making the transformation ψ(x, y) = cos(ϕ(x, y)) exp{iη(x, y)}, ψ(x, y) = cos(ϕ(x, y)) exp{−iη(x, y)}, in the complex sine-Gordon equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) yields ∇2ϕ + cos ϕ sin3 ϕ(∇η)2 − 1 2 sin(2ϕ) = 0, sin(2ϕ) ∇2η = 4∇ϕ •∇η, which is the Pohlmeyer-Lund-Regge model [33, 34, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The complex sine-Gordon equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) has a separable solution in polar coordinates given by ψ(r, θ) = Rn(r) einθ, where Rn(r) satisfies d2Rn dr2 + 1 r dRn dr + Rn 1 − R2n ��dRn dr �2 − n2 r2 � + Rn � 1 − R2 n � = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) We remark that this equation also arises in extended quantum systems [4, 5, 6], in relativity [20] and in coefficients in the three-term recurrence relation for orthogonal polynomials with respect to the weight w(θ) = et cos θ on the unit circle, see [56, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The orthogonal polynomials for this weight on the unit circle are related to unitary random matrices [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) can be shown to possess the Painlev´e property, though is not in the list of 50 equa- tions given in [25, Chapter 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) can be transformed to the fifth Painlev´e equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 9 (i) If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then making the transformation Rn(r) = 1 + un(z) 1 − un(z), r = 1 2z, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) yields d2un dz2 = � 1 2un + 1 un − 1 � �dun dz �2 − 1 z dun dz + n2(un − 1)2(u2 n − 1) 8z2un − un(un + 1) 2(un − 1) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) which is PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α = 1 8n2, β = − 1 8n2, γ = 0 and δ = − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (ii) If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then making the transformation Rn(r) = 1 � 1 − vn(x) , r = √x, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) yields d2vn dx2 = � 1 2vn + 1 vn − 1 � �dvn dx �2 − 1 x dvn dx − n2(vn − 1)2 2x2vn + vn 2x, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α = 0, β = − 1 2n2 and γ = 1 2 so is equivalent to PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This shows that solutions of equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) are related by vn(x) = 4un(z) 1 + u2n(z), x = 1 4z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The function Rn(r) satisfies the ordinary differential equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2), the differential-difference equa- tions dRn dr + n r Rn − � 1 − R2 n � Rn−1 = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7a) dRn−1 dr − n − 1 r Rn−1 + � 1 − R2 n−1 � Rn = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b) since solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7a) for Rn−1(r) and substituting in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b) yields equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Also eliminating the derivatives in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7), after letting n → n + 1 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7b), yields the difference equation Rn+1 + Rn−1 = 2n r Rn 1 − R2n , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) which is known as the discrete Painlev´e II equation [41, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If n = 1 then equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7) have the solution R0(r) = 1, R1(r) = C1I1(r) − C2K1(r) C1I0(r) + C2K0(r), where I0(r), K0(r), I1(r) and K1(r) are the imaginary Bessel functions and C1 and C2 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For solutions which are bounded at r = 0 then necesssarily C2 = 0 and so R0(r) = 1, R1(r) = I1(r) I0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) Hence one can use the difference equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) to determine Rn(r), for n ≥ 2, which yields R2(r) = −rR2 1(r) + 2R1(r) − r r [R2 1(r) − 1] , R3(r) = R3 1(r) − rR2 1(r) − 2R1(r) + r R1(r) [rR2 1(r) + R1(r) − r] , R4(r) = r(r2 + 5)R4 1(r) + 4R3 1(r) − 2r(r2 + 3)R2 1(r) + r3 r [(r2 − 1)R4 1(r) + 4rR3 1(r) − 2(r2 + 2)R2 1(r) − 4rR1(r) + r2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These results suggest that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) should be solvable in terms of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1), which is illustrated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 10 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' If Rn(r) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then wn(r) = Rn+1(r)/Rn(r) satisfies d2wn dr2 = 1 wn �dwn dr �2 − 1 r dwn dr − 2n r w2 n + 2(n + 1) r + w3 n − 1 wn , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) which is PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = −2n and β = 2(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See Hisakado [23] and Tracy & Widom [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' see also [56, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We note that since the parameters in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='10) satisfy −α + β = 4n + 2, with n ∈ Z+, then the equation has solutions expressible in terms of the modified Bessel functions I0(r) and I1(r) (as well as K0(r) and K1(r), but these are not needed here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Let τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) be the n × n determinant τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = det �� r d dr �j+k Iν(r) �n−1 j,k=0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11) with Iν(r) the modified Bessel function, then wn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) ≡ d dz � ln τn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) τn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν + 1) � − n + ν z , n ≥ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='12) satisfies PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with α = 2(ν − n) and β = 2(ν + n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' See, for example, [19, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) has the solution Rn(r) = τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) where τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) is the determinant given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The proof is straightforward using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9) we have R1(r) = I1(r) I0(r) = τ1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τ1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) is true if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Assuming (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='13) holds then from Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Rn+1(r) = wn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0)Rn(r) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) × τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) = τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) τn+1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), as required, and so the result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6) have the Bessel function solutions un(z) = τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) + τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1) − τn( 1 2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0), vn(x) = 1 − τ 2 n(√x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 0) τ 2n(√x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1), respectively, with τn(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) the determinant given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The formal asymptotic behaviour of the vortex solution Rn(r) is given by Rn(r) = rn 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' � 1 − r2 4(n + 1) + O � r4�� , as r → 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) Rn(r) = 1 − n 2r − n2 8r2 − n(n2 + 1) 16r3 + O(r−4), as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' These are determined from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2 Generalised Charlier polynomials The Charlier polynomials Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) are a family of orthogonal polynomials introduced in 1905 by Char- lier [7] given by Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 2F0 (−n, −k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' −1/z) = (−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='L(−1−k) n (−1/z) , z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='16) where 2F0(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) is the hypergeometric function and L(α) n (z) is the associated Laguerre polynomial, see, for example, [48, §18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Charlier polynomials are orthogonal on the lattice N with respect to the Poisson distribution ω(k) = zk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' , z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) and satisfy the orthogonality condition ∞ � k=0 Cm(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)Cn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)zk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ez zn δm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Smet and Van Assche [51] generalized the Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='17) with one additional parameter through the weight function ω(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = Γ(ν + 1) zk Γ(ν + k + 1) Γ(k + 1), z > 0, with ν a parameter such that ν > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' This gives the discrete weight ω(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = zk (ν + 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=', z > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) where (ν + 1)k = Γ(ν + 1 + k)/Γ(ν + 1) is the Pochhammer symbol, on the lattice N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Discrete orthogonal polynomials are characterized by the discrete Pearson equation ∆ � σ(k)ω(k) � = τ(k)ω(k), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) where ∆ is the forward difference operator ∆f(k) = f(k + 1) − f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) satisfies the discrete Pearson equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='19) with σ(k) = k(k + ν), τ(k) = −k2 − νk + z, and so the generalised Charlier polynomials are semi-classical orthogonal polynomials since τ(k) is a polynomial with deg(τ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The special case β = 0 was first considered by Hounkonnou, Hounga and Ronveaux [24] and later studied by Van Assche and Foupouagnigni [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' For the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), the orthonormal polynomials pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) satisfy the orthog- onality condition ∞ � k=0 pm(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z)pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) zk (ν + 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' = δm,n, and the three-term recurrence relation kpn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = an+1(z)pn+1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) + bn(z)pn(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) + an(z)pn−1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20) with p−1(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 0 and p0(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Our interest is in the coefficients an(z) and bn(z) in the recurrence relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Smet and Van Assche [51, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1] proved the following theorem for recurrence coefficients associated with the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 12 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence coefficients an(z) and bn(z) for orthonormal polynomials associated with the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18) on the lattice N satisfy the discrete system (a2 n+1 − z)(a2 n − z) = z(bn − n)(bn − n + ν), bn + bn−1 − n + ν + 1 = nz/a2 n, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) with initial conditions a2 0 = 0, b0 = √z Iν+1(2√z) Iν(2√z) = z d dz � ln Iν(2√z) � − ν 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='22) with Iν(k) the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The discrete system such as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) for recurrence coefficients is sometimes known as the Laguerre-Freud equations, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [3, 24, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence coefficients an(z) and bn(z) also satisfy the Toda lattice, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [56, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8] z d dz a2 n = a2 n(bn − bn−1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23a) z d dz bn = a2 n+1 − a2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23b) Letting a2 n(z) = xn(z) and bn(z) = yn(z) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23) yields (xn+1 − z)(xn − z) = t(yn − n)(yn − n + ν), z dxn dt = xn(yn − yn−1), yn + yn−1 − n + ν + 1 = nz xn , z dyn dz = xn+1 − xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Eliminating xn+1 and yn−1 in these equations yields the differential system z dxn dz = xn(2yn + ν − n + 1) − nz, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a) z dyn dz = −xn + z + (yn − n)(yn − n + ν)z xn − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) Solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a) for yn gives yn = z 2xn dxn dz + nz 2xn + n − ν − 1 2 , and substituting this into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) yields d2xn dz2 = 1 2 � 1 xn + 1 xn − z � − xn z(xn − z) dxn dz − 2x2 n z2 + 4xn + n2 − ν2 + 1 2z − n2 2xn + 1 − ν2 2(xn − z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) Making the transformation xn(z) = z 1 − wn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='26) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) yields d2wn dz2 = � 1 2wn + 1 wn − 1 ��dwn dz �2 − 1 z dwn dz + (wn − 1)2(n2w2 n − ν2) 2wnz2 − 2wn z , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='27) which is deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) with parameters α = 1 2n2, β = − 1 2ν2 and γ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Solving (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24b) for xn gives xn = − 1 2z dyn dz + z + 1 2Xn, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='28) where X2 n = z2 �dyn dz �2 + 4z(yn − n)(yn − n + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) 13 From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) we get dXn dz = z2 Xn d2yn dz2 dyn dz + z Xn �dyn dz �2 + 2z(2yn − 2n + ν) Xn dyn dz + 2(yn − n)(yn − n + ν) Xn (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='30) Substituting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='28) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='24a), then using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='30), solving for Xn, and substituting into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='29) yields the second-order, second-degree equation � 2z d2yn dz2 + dyn dz + 8yn − 8n + 4ν �2 = (4yn − 2n + 2ν + 1)2 z � z �dyn dz �2 + 4(yn − n)(yn − n + ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='31) Making the transformation yn(z) = 1 2vn(x) + 1 2n − 1 2ν − 1 4, x = 2√z, in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='31) yields �d2vn dx2 + 4vn − 4n − 2 �2 = 4v2 n x2 ��dvn dx �2 + 4v2 n − 4(2n + 1)vn + (2n + 1)2 − 4ν2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) in [14] is �d2v dx2 − av − b �2 = 4v2 x2 ��dv dx �2 − av2 − 2bv − c � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) with a, b and c parameters, an equation derived by Chazy [8], and is the primed version of equation SD-III in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Hence equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) is the special case of equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) with a = −4, b = 4n + 2, c = 4ν2 − (2n + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Cosgrove [14] showed that equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='33) is solvable in terms of solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Consequently, the solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='32) is given by vn(x) = x 2q � dq dx + q2 + 1 � , where q(x) satisfies PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for the parameters A = 2ν − 2n − 2 and B = 2ν + 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The recurrence relations an(z) and bn(z) are given by a2 n(z) = xn(z) = Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)Tn−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 n (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34a) bn(z) = yn(z) = z d dz � ln Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) where Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = det �� z d dz �j+k Iν � 2√z � �n−1 j,k=0 , with T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) = 1, and Iν(x) is the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The expression (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34a) for a2 n(z) follows immediately by substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='14) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' To prove the result (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) for bn(z) we use induction and the factor that from equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23b), a2 n(z) = xn(z) and bn(z) = yn(z) are related by z dxn dt = xn(yn − yn−1), and initially y0(z) = z d dz � ln T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � } − ν 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 14 Hence y1(z) = z d dz � ln x1(z) � + y0(z) = z d dz � ln T2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)T0(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 1 (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � + z d dz {ln T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)} − ν 2 = z d dz � ln T2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , so (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) is true for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Now suppose that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34b) is true, then yn+1(z) = z d dz � ln xn(z) � + yn(z) = z d dz � ln Tn+2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν)Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) T 2 n+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � + z d dz � ln Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 = z d dz � ln Tn+2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) Tn+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' ν) � − ν 2 , as required, and so the result follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' We remark that equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='23a) is identically satisfied by a2 n(z) and bn(z) given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' In a recent paper, Fern´andez-Irisarri and Ma˜nas [17, §2] discuss the generalised Charlier weight (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='18), in particular properties of the coefficients in the recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The relationship between the notations in [17] and those here are xn(z) = γn(η) and yn(z) = βn(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Fern´andez-Irisarri and Ma˜nas [17] relate xn(z) and yn(z) to Okamoto’s Hamiltonian for PIII′ [46] and derive two ordinary differential equations for xn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (45) in [17, Theorem 4] is the third order equation δz �xn z � δ2 z(ln xn) + 2xn � + n2z xn � = 2xn, δz(f) = z df dz , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' d3xn dz3 = 1 zx2n � z dxn dz − xn � � 2xn d2xn dz2 − �dxn dz �2 + n2 � − 4xn z2 dxn dz + 2xn(xn + z) z3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) and the state that this equation “should have the Painlev´e property”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) can be integrate to give equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25), with ν2 as the constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Since equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) is equivalent to deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) then equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='35) does have the Painlev´e property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Equation (60) in [17, Theorem 5] is the second order equation � 1 − xn z � � δz �δz(xn) + nz xn � + 2xn � + 2{xn − z + (n − b)n} = − 1 2 �δz(xn) + nz xn �2 + (n + 1) �δz(xn) + nz xn � + (n − b − 1)(3n − b + 1), which is equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='25) with ν2 = 2(b − n)2 + n2 − 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 7 Discussion In this paper the classical solutions of deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) have been classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Ohyama and Okumura [43, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1] give a list of classical solutions of PI to PV and state that “deg-P5 with α = 1 2a2, β = − 1 8, γ = −2 has the algebraic solution w(z) = 1 + 2√z/a”2 and “deg-P5 with β = 0 has the Riccati type 2As noted in [1], there is typo in [43] who say β = −8 rather than β = − 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' 15 solutions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The results in this paper show that there are more classical solutions of deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The algebraic solution is equivalent to the “seed solution” obtained by setting n = 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' u0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' µ) = µ x + µ, and there is a more general hierarchy of “Riccati type solutions” which are described in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' All solutions of PII–PVI that are expressible in terms of special functions satisfy a first-order equa- tion of the form �du dx �n = n−1 � j=0 Fj(u, x) �du dx �j , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) where Fj(u, x) is polynomial in u with coefficients that are rational functions of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It can be shown that the Bessel function solutions of PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n odd, whereas the Bessel function solutions of deg-PV (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The relationship between PIII (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) and deg-PV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) is similar to that between the second Painlev´e equation (PII) d2q dx2 = 2q3 + xq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) with α a parameter, and Painlev´e XXXIV equation (P34) d2p dx2 = 1 2p � dp dx �2 + 2p2 − xp − (α + 1 2)2 2p , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) which is equivalent to equation XXXIV of Chapter 14 in [25], in that both pairs of equations arise from a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' The Hamiltonian associated with PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) is HII(q, p, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' α) = 1 2p2 − (q2 + 1 2z)p − (α + 1 2)q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='4) and so dq dz = p − q2 − 1 2z, dp dz = 2qp + α + 1 2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='5) see [28, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It is known that PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) and P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) have special function solutions in terms of Airy functions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' It can be shown that the Airy function solutions of PII (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='2) satisfy first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n odd, whereas the Airy function solutions of P34 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='3) satisfy a first-order equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content='1) for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' Acknowledgements I thank Clare Dunning and Steffen Krusch for helpful comments and illuminating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfwv5m/content/2301.01727v1.pdf'} +page_content=' References [1] P.' 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