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Series ISSN 1939-5221
Generating Functions in Engineering and the Applied Sciences
Rajan Chattamvelli, VIT University, Vellore, Tamil Nadu
Ramalingam Shanmugam, Texas State University
This is an introductory book on generating functions (GFs) and their applications. It discusses
commonly encountered generating functions in engineering and applied sciences, such as ordinary
generating functions (OGF), exponential generating functions (EGF), probability generating functions
(PGF), etc. Some new GFs like Pochhammer generating functions for both rising and falling factorials
are introduced in Chapter 2. Two novel GFs called “mean deviation generating function” (MDGF) and
“survival function generating function” (SFGF), are introduced in Chapter 3. The mean deviation of a
variety of discrete distributions are derived using the MDGF. The last chapter discusses a large number
of applications in various disciplines including algebra, analysis of algorithms, polymer chemistry,
combinatorics, graph theory, number theory, reliability, epidemiology, bio-informatics, genetics,
management, economics, and statistics.
Some background knowledge on GFs is often assumed for courses in analysis of algorithms,
advanced data structures, digital signal processing (DSP), graph theory, etc. These are usually provided
by either a course on “discrete mathematics” or “introduction to combinatorics.” But, GFs are also used
in automata theory, bio-informatics, differential equations, DSP, number theory, physical chemistry,
reliability engineering, stochastic processes, and so on. Students of these courses may not have exposure
to discrete mathematics or combinatorics. This book is written in such a way that even those who do
not have prior knowledge can easily follow through the chapters, and apply the lessons learned in
their respective disciplines. The purpose is to give a broad exposure to commonly used techniques of
combinatorial mathematics, highlighting applications in a variety of disciplines.
ABOUT SYNTHESIS
This volume
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apply the lessons learned in
their respective disciplines. The purpose is to give a broad exposure to commonly used techniques of
combinatorial mathematics, highlighting applications in a variety of disciplines.
ABOUT SYNTHESIS
This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and
Computer Science.
research and
development topics, published quickly in digital and print formats. For more information, visit our website:
http://store.morganclaypool.com
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Generating
Functions in
Engineering and the
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Ramalingam Shanmugam
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Generating Functions
in Engineering and
the Applied Sciences
Synthesis Lectures on
Engineering
Each book in the series is written by a well known expert in the field. Most titles cover subjects
such as professional development, education, and study skills, as well as basic introductory
undergraduate material and other topics appropriate for a broader and less technical audience.
In addition, the series includes several titles written on very specific topics not covered
elsewhere in the Synthesis Digital Library.
Generating Functions in Engineering and the Applied Sciences
Rajan Chattamvelli and Ramalingam Shanmugam
2019
Transformative Teaching: A Collection of Stories of Engineering Faculty’s Pedagogical
Journeys
Nadia Kellam, Brooke Coley, and Audrey Bok
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velli and Ramalingam Shanmugam
2019
Transformative Teaching: A Collection of Stories of Engineering Faculty’s Pedagogical
Journeys
Nadia Kellam, Brooke Coley, and Audrey Boklage
2019
Ancient Hindu Science: Its Transmission and Impact of World Cultures
Alok Kumar
2019
Value Relational Engineering
Shuichi Fukuda
2018
Strategic Cost Fundamentals: for Designers, Engineers, Technologists, Estimators,
Project Managers, and Financial Analysts
Robert C. Creese
2018
Concise Introduction to Cement Chemistry and Manufacturing
Tadele Assefa Aragaw
2018
Data Mining and Market Intelligence: Implications for Decision Making
Mustapha Akinkunmi
2018
Empowering Professional Teaching in Engineering: Sustaining the Scholarship of
Teaching
John Heywood
2018
iii
The Human Side of Engineering
John Heywood
2017
Geometric Programming for Design Equation Development and Cost/Profit
Optimization (with illustrative case study problems and solutions), Third Edition
Robert C. Creese
2016
Engineering Principles in Everyday Life for Non-Engineers
Saeed Benjamin Niku
2016
A, B, See... in 3D: A Workbook to Improve 3-D Visualization Skills
Dan G. Dimitriu
2015
The Captains of Energy: Systems Dynamics from an Energy Perspective
Vincent C. Prantil and Timothy Decker
2015
Lying by Approximation: The Truth about Finite Element Analysis
Vincent C. Prantil, Christopher Papadopoulos, and Paul D. Gessler
2013
Simplified Models for Assessing Heat and Mass Transfer in Evaporative Towers
Alessandra De Angelis, Onorio Saro, Giulio Lorenzini, Stefano D’Elia, and Marco Medici
2013
The Engineering Design Challenge: A Creative Process
Charles W. Dolan
2013
The Making of Green Engineers: Sustainable Development and the Hybrid Imagination
Andrew Jamison
2013
Crafting Your Research Future: A Guide to Successful Master’s and Ph.D. Degrees in
Science & Engineering
Charles
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13
The Making of Green Engineers: Sustainable Development and the Hybrid Imagination
Andrew Jamison
2013
Crafting Your Research Future: A Guide to Successful Master’s and Ph.D. Degrees in
Science & Engineering
Charles X. Ling and Qiang Yang
2012
iv
Fundamentals of Engineering Economics and Decision Analysis
David L. Whitman and Ronald E. Terry
2012
A Little Book on Teaching: A Beginner’s Guide for Educators of Engineering and
Applied Science
Steven F. Barrett
2012
Engineering Thermodynamics and 21st Century Energy Problems: A Textbook
Companion for Student Engagement
Donna Riley
2011
MATLAB for Engineering and the Life Sciences
Joseph V. Tranquillo
2011
Systems Engineering: Building Successful Systems
Howard Eisner
2011
Fin Shape Thermal Optimization Using Bejan’s Constructal Theory
Giulio Lorenzini, Simone Moretti, and Alessandra Conti
2011
Geometric Programming for Design and Cost Optimization (with illustrative case study
problems and solutions), Second Edition
Robert C. Creese
2010
Survive and Thrive: A Guide for Untenured Faculty
Wendy C. Crone
2010
Geometric Programming for Design and Cost Optimization (with Illustrative Case Study
Problems and Solutions)
Robert C. Creese
2009
Style and Ethics of Communication in Science and Engineering
Jay D. Humphrey and Jeffrey W. Holmes
2008
v
Introduction to Engineering: A Starter’s Guide with Hands-On Analog Multimedia
Explorations
Lina J. Karam and Naji Mounsef
2008
Introduction to Engineering: A Starter’s Guide with Hands-On Digital Multimedia and
Robotics Explorations
Lina J. Karam and Naji Mounsef
2008
CAD/CAM of Sculptured Surfaces on Multi-Axis NC Machine: The DG/K-Based
Approach
Stephen P. Radzevich
2008
Tensor Properties of Solids, Part Two: Transport Properties of Solids
Richard F. Tinder
2007
Tensor Properties of Solids, Part One: Equilibrium Tensor Properties of Solids
Richard F. Tinder
2007
Essentials
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ids, Part Two: Transport Properties of Solids
Richard F. Tinder
2007
Tensor Properties of Solids, Part One: Equilibrium Tensor Properties of Solids
Richard F. Tinder
2007
Essentials of Applied Mathematics for Scientists and Engineers
Robert G. Watts
2007
Project Management for Engineering Design
Charles Lessard and Joseph Lessard
2007
Relativistic Flight Mechanics and Space Travel
Richard F. Tinder
2006
Copyright © 2019 by Morgan & Claypool
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in
any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations
in printed reviews, without the prior permission of the publisher.
Generating Functions in Engineering and the Applied Sciences
Rajan Chattamvelli and Ramalingam Shanmugam
www.morganclaypool.com
ISBN: 9781681736389
ISBN: 9781681736396
ISBN: 9781681736402
paperback
ebook
hardcover
DOI 10.2200/S00942ED1V01Y201907ENG037
A Publication in the Morgan & Claypool Publishers series
SYNTHESIS LECTURES ON ENGINEERING
Lecture #37
Series ISSN
Print 1939-5221 Electronic 1939-523X
Generating Functions
in Engineering and
the Applied Sciences
Rajan Chattamvelli
VIT University, Vellore, Tamil Nadu
Ramalingam Shanmugam
Texas State University, San Marcos, TX
SYNTHESIS LECTURES ON ENGINEERING #37
CM&cLaypoolMorganpublishers& ABSTRACT
This is an introductory book on generating functions (GFs) and their applications. It discusses
commonly encountered generating functions in engineering and applied sciences, such as or-
dinary generating functions (OGF), exponential generating functions (EGF), probability gen-
erating functions (PGF), etc. Some new GFs like Pochhammer generating functions for both
rising and falling factor
|
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|
sciences, such as or-
dinary generating functions (OGF), exponential generating functions (EGF), probability gen-
erating functions (PGF), etc. Some new GFs like Pochhammer generating functions for both
rising and falling factorials are introduced in Chapter 2. Two novel GFs called “mean deviation
generating function” (MDGF) and “survival function generating function” (SFGF), are intro-
duced in Chapter 3. The mean deviation of a variety of discrete distributions are derived using
the MDGF. The last chapter discusses a large number of applications in various disciplines in-
cluding algebra, analysis of algorithms, polymer chemistry, combinatorics, graph theory, num-
ber theory, reliability, epidemiology, bio-informatics, genetics, management, economics, and
statistics.
Some background knowledge on GFs is often assumed for courses in analysis of algo-
rithms, advanced data structures, digital signal processing (DSP), graph theory, etc. These are
usually provided by either a course on “discrete mathematics” or “introduction to combinatorics.”
But, GFs are also used in automata theory, bio-informatics, differential equations, DSP, num-
ber theory, physical chemistry, reliability engineering, stochastic processes, and so on. Students
of these courses may not have exposure to discrete mathematics or combinatorics. This book is
written in such a way that even those who do not have prior knowledge can easily follow through
the chapters, and apply the lessons learned in their respective disciplines. The purpose is to give
a broad exposure to commonly used techniques of combinatorial mathematics, highlighting ap-
plications in a variety of disciplines.
Any suggestions for improvement will be highly appreciated. Please send your comments
to [email protected], and they will be incorporated in subsequent revisions.
KEYWORDS
algebra, analysis of algorithms, bio-informatics, CDF generating functions, com-
binatorics, cumulants, difference equations, discrete mathematics, economics, epi-
demiology, finance, genetics, graph theory, management, mean deviation generat-
ing function, moments, number theory, Pochhammer generating functions, poly-
mer chemistry, power series, recurrence relations, reliability engineering, special
numbers, statistics, strided sequences, survival function, truncated distributions.
Contents
ix
List of Tables . . . . .
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, Pochhammer generating functions, poly-
mer chemistry, power series, recurrence relations, reliability engineering, special
numbers, statistics, strided sequences, survival function, truncated distributions.
Contents
ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1
Types of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Origin of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Existence of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Notations and Nomenclatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Rising and Falling Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Dummy Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3
Types of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Ordinary Generating Functions . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Ordinary Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Types of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.3 OGF for Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Exponential Generating Functions (EGF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Pochhammer Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6.1 Rising Pochhammer GF (RPGF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Falling Pochhammer GF (FPGF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6.2
1.7 Other Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.1 Auto-Covariance Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 16
Information Generating Function (IGF) . . . . .
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.7.1 Auto-Covariance Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 16
Information Generating Function (IGF) . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.2
1.7.3 Generating Functions in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.4 Generating Functions in Number Theory . . . . . . . . . . . . . . . . . . . . . . 17
1.7.5 Rook Polynomial Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.6
Stirling Numbers of Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5
1.6
1.8
2
Operations on Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1
Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Extracting Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
x
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1.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
x
2.1.3 Multiplication by Non-Zero Constant . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 Linear Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5
2.1.6
Functions of Dummy Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.7 Convolutions and Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.8 Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.9
Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2
Invertible Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Composition of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . .
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of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4
3
Generating Functions in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2
3.1 Generating Functions in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Types of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Probability Generating Functions (PGF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Properties of PGF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1
3.3 Generating Functions for CDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Generating Functions for Survival Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Generating Functions for Mean Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.6 MD of Some Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.6 MD of Some Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.1 MD of Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.2 MD of Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.3 MD of Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6.4 MD of Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Moment Generating Functions (MGF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Properties of Moment Generating Functions . . . . . . . . . . . . . . . . . . . 49
3.8 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Properties of Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.9 Cumulant Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9.1 Relations Among Moments and Cumulants . . . . . . . . . . . . . . . . . . . . 54
3.10 Factorial Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3
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. . . 54
3.10 Factorial Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Conditional Moment Generating Functions (CMGF) . . . . . . . . . . . . . . . . . . 58
3.12 Generating Functions of Truncated Distributions . . . . . . . . . . . . . . . . . . . . . . 58
3.13 Convergence of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.8.1
3.7.1
4
Applications of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xi
4.4
4.1
4.2
4.3
Applications in Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Series Involving Integer Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.1
4.1.2
Permutation Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.3 Generating Function of Strided Sequence . . . . . . . . . . . . . . . . . . . . . . 64
Applications in Computing . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . 64
Applications in Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Merge-Sort Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Quick-Sort Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.3 Binary-Search Algorithm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.4 Well-Formed Parentheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.5
Formal Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Applications in Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Combinatorial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2 New Generating Functions from Old . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.3 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.4 Towers of Hanoi Puzzle . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.4 Towers of Hanoi Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Applications in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 Graph Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.2 Tree Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Applications in Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.1
Polymer Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.2 Counting Isomers of Hydrocarbons . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Applications in Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Applications in Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
S
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Applications in Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Sums of IID Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.8.1
4.8.2
Infinite Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.8.3 Applications in Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.9 Generating Functions in Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Series-Parallel Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.10 Applications in Bioinformatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.10.1 Lifetime of Cellular Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.10.2 Sequence Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.11 Applications in Genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.12 Applications in Management . . . . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . 86
4.12 Applications in Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.12.1 Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6
4.7
4.8
4.9.1
4.5
xii
4.13 Applications in Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
List of Tables
xiii
1.1
Some standard generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Convolution as
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.1
Some standard generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Convolution as diagonal sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2
2.3
Summary of convolutions and powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Summary of generating functions operations . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1
Summary table of generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 MD of geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 MD of binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 MD of Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 MD of negative binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6
3.7
Table of characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Summary table of zero-truncated generating functions . . . . . . . . . . . . . . . . . . 59
C H A P T E R
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. . . . . . . 53
Summary table of zero-truncated generating functions . . . . . . . . . . . . . . . . . . 59
C H A P T E R 1
1
Types of Generating Functions
After finishing the chapter, readers will be able to
(cid:1) (cid:1) (cid:1)
• explain the meaning and uses of generating functions.
• describe the ordinary generating functions.
• explore exponential generating functions.
• apply generating functions to enumeration problems.
• comprehend Pochhammer generating functions.
• outline some other generating functions.
1.1
INTRODUCTION
A GF is a mathematical technique to concisely represent a known ordered sequence into a simple
algebraic function. In essence, it takes a sequence as input, and produces a continuous function
in one or more dummy (arbitrary) variables as output. The input can be real or complex num-
bers, matrices, symbols, functions, or variables including random variables. The input depends
on the area of application. For example, it is amino acid symbols or protein sequences in bio-
informatics, random variables in statistics, digital signals in signal processing, number of infected
persons in a population in epidemiology, etc. The output can contain unknown parameters, if
any.
1.1.1 ORIGIN OF GENERATING FUNCTIONS
Generating functions (GFs) were first introduced by Abraham de Moivre circa 1730, whose
work originated in enumeration. It became popular with the works of Leonhard Euler, who
used it to find the number of divisors of a positive integer (Chapter 4). The literal meaning of
enumerate is to count or reckon something. This includes persons or objects, activities, entities,
or known processes. Counting the number of occurrences of a discrete outcome is fundamental
in enumerative combinatorics. For example, it may be placing objects into urns, selecting entities
from a finite population, arranging objects linearly or circularly, etc. Thus, GFs are very useful
to explore discrete problems involving finite or infinite sequences.
2
1. TYPES OF GENERATING FUNCTIONS
Definition 1.1 (Definition of Generating Function) A GF is a simple and concise expression
in one or more dummy variables that captures the coefficients of a finite or
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GENERATING FUNCTIONS
Definition 1.1 (Definition of Generating Function) A GF is a simple and concise expression
in one or more dummy variables that captures the coefficients of a finite or infinite series, and
generates a quantity of interest using calculus or algebraic operations, or simple substitutions.
GFs are used in engineering and applied sciences to generate different quantities with
minimal work. As they convert practical problems that involve sequences into the continu-
ous domain, calculus techniques can be employed to get more insight into the problems or
sequences. They are used in quantum mechanics, mathematical chemistry, population genetics,
bio-informatics, vector differential equations, epidemiology, stochastic processes, etc., and their
applications in numerical computing include best uniform polynomial approximations, and find-
ing characteristic polynomials. Average complexity of computer algorithms are obtained using
GFs that have closed form (Chapter 4). This allows asymptotic estimates to be obtained easily.
GFs are also used in polynomial multiplications in symbolic programming languages. In statis-
tics, they are used in discrete distributions to generate probabilities, moments, cumulants, facto-
rial moments, mean deviations, etc. (Chapter 3). The partition functions, Green functions, and
Fourier series expansions are other examples from various applied science fields. They are known
by different names in these fields. For example, z-transforms (also called s-transforms) used in
digital signal processing (DSP), characteristic equations used in econometrics and numerical
computing, canonical functions used in some engineering fields, and annihilator polynomials
used in analysis of algorithms are all special types of GFs. Note, however, that the McClaurin
series in calculus that expands an arbitrary continuous differentiable function around the point
zero as f .x/
is not considered as a GF in this text be-
cause it involves derivatives of a function.
.x2=2(cid:138)/f 00.0/
xf 0.0/
C (cid:1) (cid:1) (cid:1)
f .0/
C
D
C
1.1.2 EXISTENCE OF GENERATING FUNCTIONS
GFs are much easier to work with than a whole lot of numbers or symbols. This does not mean
that any sequence can be
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f .0/
C
D
C
1.1.2 EXISTENCE OF GENERATING FUNCTIONS
GFs are much easier to work with than a whole lot of numbers or symbols. This does not mean
that any sequence can be converted into a GF. For instance, a random sequence may not have a
nice and compact GF. Nevertheless, if the sequence follows a mathematical rule (like recurrence
relation, geometric series, harmonic series), it is possible to express them compactly using a
GF. Thus, it is extensively used in counting problems where the primary aim is to count the
number of ways a task can be accomplished, if other parameters are known. A typical example
is in counting objects of a finite collection or arrangement. Explicit enumeration is applicable
only when the size (number of elements) is small. Either computing devices must be used,
or alternate mathematical techniques like combinatorics, graph theory, dynamic programming,
etc., employed when this size becomes large. In addition to the size, the enumeration problem
may also involve one or more parameters (usually discrete). This is where GFs become very
helpful. Many useful relationships among the elements of the sequence can then be derived
from the corresponding GF. If the GF has closed form, it is possible to obtain a closed form
for its coefficients as well. They are powerful tools to prove facts and relationships among the
coefficients. Although they are called generating “functions,” they are not like mathematical
functions that have a domain and range.
1.2. NOTATIONS AND NOMENCLATURES 3
2
(cid:0)
1;
1;
C
C
(cid:140)
(cid:0)
Convergence of Generating Functions
Convergence of the GF is assumed in some fields for limited values of the parameters or
dummy variables. For example, GF associated with real-valued random variables assumes that
the dummy variable t (defined below) is in the range (cid:140)0; 1(cid:141), whereas for complex-valued ran-
1(cid:141). Similarly, the GF used in signal processing assumes that
dom variables the range is (cid:140)
t
1(cid:141). As the input can be complex numbers in some signal processing applications, the
GF can also be complex. A formal power series implies that nothing is assumed on the
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range is (cid:140)
t
1(cid:141). As the input can be complex numbers in some signal processing applications, the
GF can also be complex. A formal power series implies that nothing is assumed on the conver-
gence. As shown below, the sequence 1; x; x2; x3; : : : has GF F(x) = 1=.1
x/ which converges
x2/,
in the region 0 < x < 1, while a sub-sequence 1; 0; x2; 0; x4; : : : has GF G(x) = 1=.1
1 < x < 1. The region within which it converges is called the Radius
which converges for
of Convergence (RoC). Thus the RoC of a series is the set of bounded values within which
it will converge. Obviously, .1; 1; : : :/ (infinite series) has OGF 1=.1
x/, so that the RoC is
cx/ so that the RoC is 0 < x < 1=c. An
0 < x < 1. Similarly, .c; c; c; : : :/ is generated by 1=.1
OGF being weighted by xk will converge if all terms beyond a value (say R) is upper-bounded.
Such a value of R, which is a finite real number, is called its RoC. This is given by the ratio
1=R
, where vertical bars denote absolute value for real coefficients. It can
easily be found by taking the ratio of two successive terms of the series, if the nth term is ex-
pressed as a simple mathematical function. Convergence of a GF is often ignored in this book
because it is rarely evaluated if at all, and that too for dummy variable values 0 or 1 (an exception
appears in Chapter 4, page 86).
1=an
!1j
Ltn
an
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
C
j
1.2 NOTATIONS AND NOMENCLATURES
A finite sequence will be denoted by placing square-brackets around them, assuming that each
element is distinguishable. An English alphabet will be used to identify it when necessary
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1.2 NOTATIONS AND NOMENCLATURES
A finite sequence will be denoted by placing square-brackets around them, assuming that each
element is distinguishable. An English alphabet will be used to identify it when necessary. An
infinite series will be denoted by simple brackets, and named using Greek alphabet. Thus, S
D
.a0; a1; a2; a3; : : : an; : : :/ denotes an
(cid:140)a0; a1; a2; a3; a4(cid:141) denotes a finite sequence, whereas (cid:27)
infinite sequence. The summation sign (P) will denote the sum of the elements that follow. The
0 pkt k
index of summation will either be explicitly specified, or denoted implicitly. Thus, P1k
D
0 pkt k implicitly assumes
is an explicit summation in which k varies from 0–
(cid:21)
that the summation is varying from 0 to an upper limit (which will be clear from context).
An assumption that n is a power of 2 may simplify the derivation in some cases (see Chapter 4,
pp. 65–66). Random variables will be denoted by uppercase letters, and particular values of them
by lowercase letters.
, whereas Pk
1
D
4
1. TYPES OF GENERATING FUNCTIONS
1.2.1 RISING AND FALLING FACTORIALS
This section introduces a particular type of the product form presented above. These expres-
sions are useful in finding factorial moments of discrete distributions, whose probability mass
function (PMF) involves factorials or binomial coefficients. In the literature, these are known
as Pochhammer’s notation for rising and falling factorials. This will be explored in subsequent
chapters.
1. Rising Factorial Notation
Factorial products come in two flavors. In the rising factorial, a variable is incremented
successively in each iteration. This is denoted as
x.j /
.x
x
(cid:3)
C
1/
D
(cid:3) (cid:1) (cid:1) (cid:1) (cid:3)
.x
j
C
(cid:0)
1/
D
j
1
(cid:
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C
1/
D
(cid:3) (cid:1) (cid:1) (cid:1) (cid:3)
.x
j
C
(cid:0)
1/
D
j
1
(cid:0)
Y
.x
0
k
D
k/
C
D
(cid:128).x
j /
C
(cid:128).x/
:
(1.1)
2. Falling Factorial Notation
In the falling factorial, a variable is decremented successively at each iteration. This is
denoted as
x.j /
.x
x
(cid:3)
(cid:0)
1/
D
(cid:3) (cid:1) (cid:1) (cid:1) (cid:3)
.x
j
(cid:0)
C
1/
D
j
1
(cid:0)
Y
0
k
D
.x
k/
(cid:0)
D
.x
x(cid:138)
j /(cid:138) D
(cid:0)
j (cid:138)
!
:
x
j
(1.2)
Writing Equation (1.1) in reverse gives us the relationship x.j /
writing Equation (1.2) in reverse gives us the relationship x.j /
D
.x
.x
j
D
(cid:0)
1/.j /. Similarly,
j
(cid:0)
1/.j /.
C
C
1.2.2 DUMMY VARIABLE
An ordinary GF (OGF) is denoted by G.t/, where t is the dummy (arbitrary) variable. It can
be any variable you like, and it depends on the field of application. Quite often, t is used as
dummy variable in statistics, s in signal processing, x in genetics, bio-informatics, and orthog-
onal polynomials, z in analysis of algorithms, and L in auto-correlation, and time-series analy-
sis.1 An exponential GF is denoted by H.t /. As GFs in some fields are associated with variables
(like random variables in statistics), they may appear before the dummy variable. For example,
G.x; t/ denotes an OGF of a random variable X. Some authors use x as a subscript as Gx.t/.
These dummy variables assume special values (like 0 or 1) to generate
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the dummy variable. For example,
G.x; t/ denotes an OGF of a random variable X. Some authors use x as a subscript as Gx.t/.
These dummy variables assume special values (like 0 or 1) to generate a quantity of interest.
Thus, uniform and absolute convergence at these points are assumed. The GFs used in statistics
can be finite or infinite, because they are defined on (sample spaces of ) random variables, or
1Analysis of algorithms is used to either select an algorithm with minimal execution time from a set of possible choices, or
minimum storage or network communication requirements. As the attribute of interest is time, the recurrences are developed
as T .n/ where n is the problem size. This is one of the reasons for choosing a dummy variable other than t.
1.3. TYPES OF GENERATING FUNCTIONS 5
on a sequence of random variables. Note that the GF is also defined for bivariate and multi-
variate data. Bivariate GF has two dummy variables. This book focuses only on univariate GF.
Although the majority of GFs encountered in this book are of “sum-type” (additive GF), there
also exist GFs of “product-type.” One example is the GF for the number of partitions of an
integer (Chapter 4). These are called multiplicative GF. The GF for the number of partitions of
n into distinct parts is p.t/
t n/.
Q1n
D
D
1.1
C
1.3 TYPES OF GENERATING FUNCTIONS
GFs allow us to describe a sequence in as simple a form as possible. Thus, a GF encodes a
sequence in compact form, instead of giving the nth term as output. Depending on what we wish
to generate, there are different GFs. For example, moment generating function (MGF) generates
moments of a population, and probability generating function (PGF) generates corresponding
probabilities. These are specific to each distribution. An advantage is that if the MGF of an
arbitrary random variable X is known, the MGF of any linear combination of the form a
b can be derived easily. This reasoning holds for other GFs too.
C
X
(cid:3)
1.4 ORDINARY GENERATING FUNCTIONS
Let (cid:27)
D
.a0; a1; a2; a3;
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derived easily. This reasoning holds for other GFs too.
C
X
(cid:3)
1.4 ORDINARY GENERATING FUNCTIONS
Let (cid:27)
D
.a0; a1; a2; a3; : : : an; : : :/ be a sequence of bounded numbers. Then the power series
f .x/
D
anxn
1
X
0
n
D
(1.3)
is called the Ordinary Generating Function (OGF) of (cid:27). Here x is a dummy variable, n is
the indexing variable (indexvar) and a0ns are known constants. For different values of an, we
get different OGF. The set of all possible values of the indexing variable of the summation in
(1.3) is called the “index set.” If the sequence is finite and of size n, we have a polynomial
of degree n. Otherwise it is a power series. Such a power series has an inherently powerful
enumerative property. As they lend themselves to algebraic manipulations, several useful and
interesting results can be derived from them as shown in Chapter 2. There exist a one-to-one
correspondence between a sequence, and its GF. This can be represented as
.a0; a1; a2; a3; : : : ; an; : : :/
(cid:148)
anxn:
1
X
0
n
D
(1.4)
We get f .x/
1 when all an
(cid:0)
1 for n even. These are represented as
x/(cid:0)
.1
D
D
an
D C
1, and .1
C
x/(cid:0)
1 when an
D (cid:0)
1 for n odd, and
.1; 1; 1; 1; : : : ; 1; : : :/
1; : : :/
1; : : : ; 1;
(cid:0)
(cid:148)
(cid:148)
.1
.1
(cid:0)
C
1
x/(cid:0)
1:
x/(cid:0)
1; 1;
.1;
(cid:0)
(cid:0)
6
(cid:0)
1/=.x
(cid:0)
1
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1
x/(cid:0)
1:
x/(cid:0)
1; 1;
.1;
(cid:0)
(cid:0)
6
(cid:0)
1/=.x
(cid:0)
1, and odd coefficients a2n
1. TYPES OF GENERATING FUNCTIONS
The OGF is .xn
1/=.x
1/ when there are a finite number of 1’s (say n of them). Thus,
(cid:0)
.1; 1; 1; 1; 1/ has OGF .x5
1 when even coefficients
x2/(cid:0)
0. The OGF is called the McClaurin series of the
a2n
function on the left-hand side (LHS) when the right-hand side (RHS) has functional values as
coefficients. In the particular case when the sum of all the coefficients is 1, it is called a PGF,
which is discussed at length in Chapter 3.
1/. Similarly, we get .1
D C
D
(cid:0)
(cid:0)
C
1
Consider the set of all positive integers P
3x2
4x3
1
. We know that 1=.1
C
x/2
sides with respect to (w.r.t.) x to get
the GF is 1=.1
(cid:0)
(cid:0)
x/2 or equivalently .1
C (cid:1) (cid:1) (cid:1)
x/
C (cid:1) (cid:1) (cid:1)
C
D
3x2
1=.1
C
D
(cid:0)
2. This can be represented as
x/(cid:0)
C
.
(cid:0)
C
C
x4
2x
x2
1/
x3
1
D
x
C
(cid:3)
(cid:0)
(cid:0)
C
C
. Differentiate both
. Thus,
4x3
C
C (cid:1) (cid:1) (cid:1)
.1; 2; 3; 4; 5; : : :/. This has OGF 1
2x
.1;
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C (cid:1) (cid:1) (cid:1)
.1; 2; 3; 4; 5; : : :/. This has OGF 1
2x
.1; 2; 3; 4; : : : ; n; : : :/
(cid:148)
.1
.1
2
(cid:0)
x/(cid:0)
2
x/(cid:0)
1=.1
1=.1
(cid:0)
x/2
x/2:
C
D
D
(cid:148)
C
2; 3;
.1;
(cid:0)
(cid:0)
4; : : : ; n;
.n
(cid:0)
C
1/; : : :/
See Table 1.1 for more examples.
Table 1.1: Some standard generating functions
FunctionSeriesTypeNotation1 ± x1 ± xOGF(1,±1,0,0,…)1 + 2x + 3x21 + 2x + 3x2OGF(1,2,3,0,0,…)11 – x∞ xkOGF(1,1,1,1,…)11 + x∞ (–1)kxkOGF(1,-1,1,-1,…)11 – 2x∞ 2kxkOGF(1,2,4,8,…)1(1 – x)2∞ (k + 1)xkOGF(1,2,3,4,…)11 – ax∞ ak xkOGF(1,a, a2, a3,…)11 – x2∞ x2kOGF(1, 0, 1, 0,…)exp(x)∞ xk/k!EGF(1, 1/1!, 1/2!, 1/3!…)exp(ax)∞ (ax)k/k!EGF(1, a/1!, a2/2!, a3/3!,…)k=0k=0k=0k=0k=0k=0k=0k=0 1.4. ORDINARY GENERATING FUNCTIONS 7
1.4.1 RECURRENCE RELATIONS
There are in general two
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=0k=0k=0k=0k=0k=0k=0 1.4. ORDINARY GENERATING FUNCTIONS 7
1.4.1 RECURRENCE RELATIONS
There are in general two ways to represent the nth term of a sequence: (i) as a function of n, and
(ii) as a recurrence relation between nth term and one or more of the previous terms. For example,
an
.1
and Jones, 1996].
1=n gives an explicit expression for nth term. Taking the ratio an=an
(cid:0)
1=n/ gives an
D
1, which is a first-order recurrence relation [Chattamvelli
1=n/an
D
(cid:0)
1/=n
.n
.1
D
D
(cid:0)
(cid:0)
(cid:0)
1
A recurrence relation can be defined either for a sequence, or on the parameters of a func-
tion (like a density function in statistics). It expresses the nth term in terms of one or more prior
1/th term. It
terms. First-order recurrence relations are those in which nth term is related to .n
is called a linear recurrence relation if this relation is linear. First-order linear recurrence relations
are easy to solve, and it requires one initial condition. A second-order recurrence relation is one
in which the nth term is related to .n
2/th terms. It needs two initial conditions
to ensure that the ensuing sequence is unique. One classical example is the Fibonacci numbers.
In general, an nth-order recurrence relation expresses the nth term in terms of n
1 previous
1
terms. Consider the Bell numbers defined as Bn
k (cid:1)Bk. This generates 1, 1, 2, 5, 15,
(cid:0)
52, 203, etc. Recurrence relations are extensively discussed in Chapter 4 (page 71).
1/th and .n
1
0 (cid:0)
P
(cid:0)
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
n
k
n
1.4.2 TYPES OF SEQUENCES
(cid:0)
There are many types of sequences encountered in engineering. Examples are arithmetic se-
quence, geometric sequence, exponential sequence, Harmonic sequence
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:0)
n
k
n
1.4.2 TYPES OF SEQUENCES
(cid:0)
There are many types of sequences encountered in engineering. Examples are arithmetic se-
quence, geometric sequence, exponential sequence, Harmonic sequence, etc. Alternating se-
quence is one in which the sign of the coefficients alter between plus and minus. This is of
a4; : : :). The sum of an alternating series may be convergent or di-
the form (a1;
a2; a3;
(cid:0)
1/n
1=n to get the alternating harmonic series
1.
vergent. Consider P1n
(cid:0)
D
, which is convergent as the sum is log.2/.
1
1=2
1.
P1n
C
(cid:0)
(cid:0)
D
1=.2n
Similarly, P1n
D
1=4
C
1/ converges to (cid:25)=4.
1an. Take an
1=3
1.1=n/
1.
D
C (cid:1) (cid:1) (cid:1)
D
1/n
C
A telescopic series is one in which all terms of its partial sum cancels out except the first or
. Use partial fractions to
1/ so that when the upper limit of the summation is finite,
1 1=(cid:140)n.n
last (or both). Consider P1n
D
write 1=(cid:140)n.n
1=.n
all terms cancel out except the first and last.
C (cid:1) (cid:1) (cid:1)
1=12
1=n
1=2
1/n
1=6
1/(cid:141)
1/(cid:141)
C
C
C
C
D
C
C
D
(cid:0)
(cid:0)
(cid:0)
C
Consider the geometric sequence (cid:140)P; Pr; Pr2; : : : ; Prn(cid:141). It is called a geometric progres-
sion in mathematics and geometric sequence in engineering. It also appears in various physical,
natural, and life sciences. As examples, the equations of state of gaseous mixtures (like
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141). It is called a geometric progres-
sion in mathematics and geometric sequence in engineering. It also appears in various physical,
natural, and life sciences. As examples, the equations of state of gaseous mixtures (like van der
Vaals equation, Virial equation), mean-energy modeling of oscillators used in crystallography
and other fields, and geometric growth and deprecation models used in banking and finance can
all be modeled as geometric series. Such series also appear in population genetics, insurance,
exponential smoothing models in time-series, etc. Here the first term is P , and rate of progres-
sion (called common ratio) is r. It is possible to represent the sum of any k terms in terms of 3
8
1. TYPES OF GENERATING FUNCTIONS
1ak
variables, P (initial term), r (common ratio), and k. An interesting property of the geometric
a2
k, i.e., the square of kth term is the product of the two terms
progression is that ak
around it. A geometric series can be finite or infinite. It is called an alternating geometric series
(cid:140)1; r; r 2; r 3; : : : ; r n(cid:141) denote a finite geometric sequence.
if the signs differ alternatively. Let Sn
D
n
1/. Both the numerator and denominator are nega-
1/=.r
Then the sum P
k
r/.
tive when 0 < r < 1, so that we could write the RHS in the alternate form .1
1/=.1
0 r k
.r n
r n
D
D
(cid:0)
(cid:0)
C
C
D
(cid:0)
1
1
C
(cid:0)
(cid:0)
Example 1.1 (OGF of alternating geometric sequence) Find the OGF of alternating geo-
metric sequence .1;
a3; a4; : : :/.
a; a2;
(cid:0)
(cid:0)
Solution 1.1 Consider the infinite series expansion of .1
C
, which is the OGF of the given sequence. Here “a” is any nonzero constant.
D
C
(cid:0)
1
ax
1
ax/(
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1 Consider the infinite series expansion of .1
C
, which is the OGF of the given sequence. Here “a” is any nonzero constant.
D
C
(cid:0)
1
ax
1
ax/(cid:0)
a2x2
a3x3
(cid:0)
C
(cid:1) (cid:1) (cid:1)
Example 1.2 (Find the OGF of an) Find the OGF of (i) an
1=3n.
D
2n
C
3n and (ii) an
2n
C
D
Solution 1.2 The OGF is G.x/
the closed form formula for geometric series to get G.x/
5x/=(cid:140).1
G.x/
x=3/ which is the required OGF.
P1n
D
(cid:0)
1=.1
2x/.1
(cid:0)
2x/
1=.1
C
D
0.2n
D
(cid:0)
C
(cid:0)
(cid:0)
3x/(cid:141) which is the required GF. In the second case a similar procedure yields
D
D
C
(cid:0)
(cid:0)
3n/xn. Split this into two terms and use
2x/
1=.1
1=.1
3x/
.2
Example 1.3 (OGF of odd positive integers) Find the OGF of the sequence 1, 3, 5, 7, 9, 11,...
Solution 1.3 The nth term of the sequence is obviously .2n
0.2n
1/t n. Split this into two terms, and take the constant 2 outside the summation to get G.t/
0 nt n
2 P1n
t/ so that G.t/
D
t/2 as a common denominator and simplify to get G.t/
t/2
2t=.1
(cid:0)
t/=.1
.1
P1n
C
D
1=.1
C
(cid:0)
t/2 as the required OGF.
(cid:0)
0 t n. The first sum is 2t=.1
t/2 and second one is 1=.1
1/. Write G.t /
t
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0)
t/2 as the required OGF.
(cid:0)
0 t n. The first sum is 2t=.1
t/2 and second one is 1=.1
1/. Write G.t /
t/. Take .1
P1n
D
C
D
C
(cid:0)
(cid:0)
(cid:0)
C
D
D
D
Example 1.4 (OGF of perfect squares) Find the OGF for all positive perfect
.1; 4; 9; 16; 25; : : :/.
squares
Solution 1.4 Consider the expression x=.1
series expansion
(cid:0)
x/2. From pages 2–3 we see that this is the infinite
D
Differentiate both sides w.r.t. x to get
(cid:0)
x=.1
x/2
x
2x2
C
C
3x3
C (cid:1) (cid:1) (cid:1) C
nxn
:
C (cid:1) (cid:1) (cid:1)
.@=@x/ (cid:0)x=.1
x/2(cid:1)
1
C
D
22x
C
(cid:0)
32x2
C (cid:1) (cid:1) (cid:1) C
n2xn
(cid:0)
1
:
C (cid:1) (cid:1) (cid:1)
(1.5)
(1.6)
If we multiply both sides by x again, we see that the coefficient of xn on the RHS is n2. Hence,
x @
x/2,
take log and differentiate to get2
x/2/ is the GF that we are looking for. To find the derivative of f
@x .x=.1
x=.1
D
(cid:0)
(cid:0)
1.4. ORDINARY GENERATING FUNCTIONS 9
.1=f /
@f
@x D
1=x
2=.1
x/
(cid:0)
D
.1
C
C
x/=(cid:140)x.1
x/(cid:141)
(cid:0)
(1.7)
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=x
2=.1
x/
(cid:0)
D
.1
C
C
x/=(cid:140)x.1
x/(cid:141)
(cid:0)
(1.7)
from which @f
C
perfect squares of all positive integers as x.1
as follows:
@x D
x/=.1
.1
(cid:0)
x/3. Multiply this expression by x to get the desired GF of
x/3. This can be represented symbolically
x/=.1
C
(cid:0)
.1; 1; 1; 1; : : : ; n; : : :/
Derivative:
.1; 2; 3; 4; : : : ; n; : : :/
Rightshift:
.0; 1; 2; 3; 4; : : : ; n; : : :/
(cid:148)
(cid:0)
D
.1
.1
1
x/(cid:0)
x/(cid:0)
(cid:148)
1=.1
(cid:0)
x/
x/2
2
1=.1
D
2
x/(cid:0)
(cid:0)
x=.1
Derivative:
.1; 4; 9; 16; : : : ; n2; : : :/
Rightshift:
.0; 1; 4; 9; 16; : : : ; n2; : : :/
(cid:148)
x.1
C
(cid:0)
x.1
@f
@x
(cid:148)
(cid:148)
(cid:0)
.x=.1
D
x/2/
(cid:0)
x/=.1
D
x/3:
(cid:0)
(cid:0)
.1
x/2
x/=.1
x/3
(cid:0)
C
(1.8)
Example 1.5 (OGF of even sequence) Find OGF of the sequence (2, 4, 10, 28, 82,: : :).
Solution 1.5 Suppose we subtract 1 from each number in the sequence. Then we get (1, 3, 9,
27, 81, 2
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, 28, 82,: : :).
Solution 1.5 Suppose we subtract 1 from each number in the sequence. Then we get (1, 3, 9,
27, 81, 243, : : :), which are powers of 3. Hence, above sequence is the sum of .1; 1; 1; 1; : : :/
.1; 3; 32; 33; 34; : : :/ with respective OGFs 1=.1
3x/. Add them to get 1=.1
x/
3x/ as the answer.
x/ and 1=.1
C
(cid:0)
1=.1
(cid:0)
(cid:0)
C
(cid:0)
1.4.3 OGF FOR PARTIAL SUMS
If the GF for any sequence is known in compact form (say F .x/), the GF for the sum of the
first n terms of that particular sequence is given by F .x/=.1
x/. A consequence of this is that
if we know the OGF of the sum of the first n terms of a sequence, we could get the original
sequence by multiplying it by .1
x/ is the same as dividing by
1=.1
x/ we could state this as follows:
x/. As multiplying by .1
(cid:0)
(cid:0)
(cid:0)
(cid:0)
Theorem 1.1 (OGF of G.x/=.1
of the partial sums of coefficients, and dividing by 1=.1
x/) Multiplying an OGF by 1=.1
(cid:0)
(cid:0)
x/ results in differenced sequence.
x/ results in the OGF
(cid:0)
Proof. Let G.x/ be the OGF of the infinite sequence (a0; a1; : : : ; an; : : :). By definition G.x/
a0
1 as a power series 1
. Expand .1
anxn
a2x2
a1x
x2
x
x/(cid:0)
C
C
2Note that the nth derivative of 1=.1
C (cid:1) (cid:1) (cid:1) C
C
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a1x
x2
x
x/(cid:0)
C
C
2Note that the nth derivative of 1=.1
C (cid:1) (cid:1) (cid:1) C
C (cid:1) (cid:1) (cid:1)
(cid:0)
x/n
C
x/ is n(cid:138)=.1
(cid:0)
(cid:0)
1 and that of 1=.1
ax/b is .n
b
(cid:0)
C
(cid:0)
C
C
D
,
C (cid:1) (cid:1) (cid:1)
1/(cid:138)an=(cid:140).b
(cid:0)
1/(cid:138).1
(cid:0)
ax/b
n(cid:141).
C
10
1. TYPES OF GENERATING FUNCTIONS
and multiply with G.x/ to get the RHS as g.x/
.1
x
C
C
x2
C
x3
C (cid:1) (cid:1) (cid:1)
/.a0
a1x
C
Change the order of summation to get
.1
D
(cid:0)
a2x2
0
C
1
x/(cid:0)
(cid:3)
G.x/
D
C (cid:1) (cid:1) (cid:1) C
1
X
@
0
j
1:xj
A
D
anxn
1
1
X
0
k
D
/
C (cid:1) (cid:1) (cid:1)
D
ak:xk!
:
(1.9)
0
@
1
X
0
k
D
0
k
X
@
0
j
D
1
aj :1
A
xk
1
A D
1
X
0
k
D
0
k
X
@
0
j
D
This is the OGF of the given sequence.
1
A
aj
xk
a0
C
D
.a0
C
a1/x
.a0
a1
C
C
C
a2/x2
:
C (cid:1) (
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1
A
aj
xk
a0
C
D
.a0
C
a1/x
.a0
a1
C
C
C
a2/x2
:
C (cid:1) (cid:1) (cid:1)
(1.10)
(cid:3)
This result has great implications. Computing the sum of several terms of a sequence
appears in several fields of applied sciences. If the GF of such a sequence has closed form, it
is a simple matter of multiplying the GF by 1=.1
x/ to get the new GF for the partial sum;
x gives the OGF of differences. Assume that the OGF of a sequence an
and multiplying by 1
is known in closed form. We seek the OGF of another sequence dn
1. As this is a
(cid:0)
x/A.x/, where
1 dk
telescopic series, P
A.x/ is the OGF of an’s.
a0. It follows that D.x/
D
Pn dnxn
(cid:0)
.1
an
an
an
D
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
n
k
D
.a0; a1; a2; a3; : : : ; an; : : :/
F .x/
(cid:148)
a2; : : :/
.a0; a0
a1; a0
a1
F .x/=.1
x/:
(1.11)
C
C
n
Consider the problem of finding the sum of squares of the first n natural numbers as P
k
D
The OGF of n2 was found in Example 1.4 as x.1
x/3. If this is divided by .1
(cid:0)
n
we should have the OGF of P
k
1 k2. This gives the OGF as x.1
x/=.1
x/=.1
x/4.
(cid:148)
C
C
(cid:0)
(cid:0)
1 k2.
x/,
D
C
(cid:0)
Example 1.6 (OGF of harmonic series) Find the OGF of 1
1=2
1
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C
(cid:0)
(cid:0)
1 k2.
x/,
D
C
(cid:0)
Example 1.6 (OGF of harmonic series) Find the OGF of 1
1=2
1=3
C
C
C (cid:1) (cid:1) (cid:1) C
1=n.
Solution 1.6 We know that
x2=2
x/
(cid:0)
1 xk=k. We also know that when an OGF is multiplied by 1=.1
x3=3
C
C
x/, we get a new
(cid:1) (cid:1) (cid:1) D
OGF in which the coefficients are the partial sum of the coefficients of the original one. Thus,
the above sequence has OGF
log.1=.1
P1k
D
C
(cid:0)
x4=4
x/=.1
log.1
log.1
x//
x/.
D
D
C
(cid:0)
(cid:0)
x
(cid:0)
(cid:0)
(cid:0)
As another example, consider the sequence .1=2;
form the convolution of log.1
1=2/
1=3/
C
x/(cid:141)2. Similarly, .1=2/(cid:140)log.1
x3.1
1=2
C
C
x/ and 1=.1
/. First
x2.1
C
. Now integrate both sides to get the OGF as .1=2/(cid:140)log.1
1
C
(cid:0)
x/ to get log.1
4 .1
x/=.1
C
C
(cid:1) (cid:1) (cid:1)
(cid:0)
C
C
C
D
C
1
2 C
x/
1
3 /;
x
1
2 /; 1
1
3 .1
x/(cid:141)2 is the OGF of .1=2; 1
3 .1
C
2 /; 1
4 .1
1
2 C
1
3 /;
(cid:1) (cid:1) (cid
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is the OGF of .1=2; 1
3 .1
C
2 /; 1
4 .1
1
2 C
1
3 /;
(cid:1) (cid:1) (cid:1)
/.
C
C (cid:1) (cid:1) (cid:1)
(cid:0)
1.5. EXPONENTIAL GENERATING FUNCTIONS (EGF)
11
m
k
1/k(cid:0)
A direct application of the above result is in proving the combinatorial
0.
P
According to the above theorem, the sum of the first m terms has OGF .1
out .1
(cid:0)
1, which is the OGF of the RHS. This proves the result.
n
1
m (cid:1). First consider the sequence .
(cid:0)
n
k(cid:1), which has OGF .1
x/n=.1
x/ to get .1
1/m(cid:0)
1/k(cid:0)
identity
x/n.
(cid:0)
x/. Cancel
n
k(cid:1)
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
D
.
x/n
(cid:0)
(cid:0)
(cid:0)
D
D
F .
.F .t/
0 akt k, then G.t/
Theorem 1.2 If F .t/
P1k
D
Proof. The proof follows easily from the fact that F .
when t index is odd so that it cancels out when we sum both. Similarly, G.t/
F .
(cid:0)
1
2 .F .t/
C
m=2
0 a2k
Pb
c
k
D
If F .t/ is an OGF, .1
(cid:0)
D
1
2 .F .t/
t// is the OGF of
x (see Chapter 4, § 4.1.1, pp. 62).
(cid:3)
.F .t/
0 akt k where the upper limit is m, then G.t/
C
t/ has coefficients with negative sign
0 a2kt 2k, and G.t/
denotes the greatest integer
1. If F .t/
1t 2k
C
m=2
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t/ has coefficients with negative sign
0 a2kt 2k, and G.t/
denotes the greatest integer
1. If F .t/
1t 2k
C
m=2
t// is the GF of Pb
c
k
t//=2 is the GF of P1k
D
k(cid:1)F .t/k is also a GF.
x
b
c
F .t//n
0 a2kt 2k.
1. Here
P1k
D
(cid:0)
1t 2k
m
P
k
D
(cid:20)
0 a2k
t//=2
D
F .
F .
0 (cid:0)
P
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
n
k
C
C
D
C
D
n
C
D
D
1.5 EXPONENTIAL GENERATING FUNCTIONS (EGF)
The primary reason for using a GF in applied scientific problems is that it represents an entire
sequence as a single mathematical function in dummy variables, which is easy to investigate and
manipulate algebraically. Whereas an OGF is used to count coefficients involving combina-
tions, an EGF is used when the coefficients are permutations or functions thereof. The EGF is
preferred to OGF in those applications in which order matters. Examples are counting strings
of fixed width formed using binary digits .0; 1/. Although bit strings 100 and 010 contain two
0’s and one 1, they are distinct.
Definition 1.2 The function
H.x/
D
1
X
0
n
anxn=n(cid:138)
(1.12)
D
is called the exponential generating function (EGF). There are two ways to interpret the divisor
of nth term. Writing it as
H.x/
D
1
X
0
n
D
bnxn where
bn
an=n(cid:138)
D
gives an OGF. But it is the usual practice to write it either as (1.12) or as
H.x/
D
1
X
0
n
D
anenxn where en
1
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D
gives an OGF. But it is the usual practice to write it either as (1.12) or as
H.x/
D
1
X
0
n
D
anenxn where en
1=n(cid:138)
D
(1.13)
(1.14)
where we have two independent multipliers. As the coefficients of exponential function .exp.x//
are reciprocals of factorials of natural numbers, they are most suited when terms of a sequence
12
1. TYPES OF GENERATING FUNCTIONS
grow very fast. EGF is a convenient way to manipulate such infinite series. Consider the expan-
sion of ex as
akxk;
ex
D
1
X
0
k
D
(1.15)
where ak
.1; 1=2(cid:138); 1=3(cid:138); : : :/. If k(cid:138) is always associated with xk, we could alternately write the above as
1=k(cid:138). This can be considered as the OGF of the sequence 1=k(cid:138), namely the sequence
D
akxk=k(cid:138);
ex
D
1
X
0
k
D
(1.16)
where ak
D
1 for all values of k.
As the denominator n(cid:138) grows fast for increasing values of n, it can be used to scale down
functions that grow too fast. For example, number of permutations, number of subsets of an
n-element set which has the form 2n, etc., when multiplied by xn=n(cid:138) will often result in a con-
verging sequence.3 Thus, e2x can be considered as the GF of the number of subsets.
These coefficients grow at different rates in practical applications. If this growth is too
fast, the multiplier xn=n(cid:138) will often ensure that the overall series is convergent. Quite often,
both the OGF and EGF may exist for a sequence. As an example, consider the sequence
1; 2; 4; 8; 16; 32; : : : with nth term an
2x/, whereas the EGF is
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GF may exist for a sequence. As an example, consider the sequence
1; 2; 4; 8; 16; 32; : : : with nth term an
2x/, whereas the EGF is e2x.
Thus, the EGF may converge in lot many problems where the OGF either is not convergent or
k(cid:1)xk. We could write this as an EGF by expanding
does not exist. Consider .1
n
(cid:0)
k(cid:1)
2n. The OGF is 1=.1
n(cid:138)=(cid:140)k(cid:138).n
k/(cid:138)(cid:141) as
x/n
0 (cid:0)
P
D
D
C
(cid:0)
n
k
D
n
D
(cid:0)
x/n
.1
C
D
n(cid:138)=.n
(cid:0)
k/(cid:138) .xk=k(cid:138)/
(1.17)
n
X
0
k
D
so that ak
n(cid:138)=.n
taken k at a time).
D
(cid:0)
k/(cid:138) (which happens to be P .n; k/, the number of permutations of n things
From Equation (1.13) it is clear that the EGF can be obtained from OGF by replacing
every occurrence of t by et and vice versa. It was shown before that the OGF of .1; 1; 1; : : :/ is
x/. Multiply the numerator and denominator by n(cid:138) to
1=.1
x/. As
get P1n
D
the number of permutations of size n is n(cid:138), this is the GF for permutations.
D
x/. This shows that the EGF of .1; 2(cid:138); 3(cid:138); : : :/ is 1=.1
x/ so that P1n
D
xn=n(cid:138)
D
(cid:2)
1=.1
1=.1
0 n(cid:138)
0 1
xn
(cid:0)
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x/ so that P1n
D
xn=n(cid:138)
D
(cid:2)
1=.1
1=.1
0 n(cid:138)
0 1
xn
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:2)
Now consider
(cid:16)ebt
(cid:0)
eat (cid:17) =.b
a/
(cid:0)
D
t=1(cid:138)
.b2
a2/=.b
a/t 2=2(cid:138)
C
.bn
(cid:0)
an/=.b
(cid:0)
a/t n=n(cid:138)
.b3
(cid:0)
a3/=.b
(cid:0)
a/t 3=3(cid:138)
C (cid:1) (cid:1) (cid:1)
(1.18)
C
:
3The sequence Pn
(cid:21)
C
0 n(cid:138)xn converges only at x
(cid:0)
(cid:0)
0.
D
C (cid:1) (cid:1) (cid:1)
1.5. EXPONENTIAL GENERATING FUNCTIONS (EGF)
13
The general solution to the Fibonacci recurrence is of the form .(cid:12)n
(cid:11)/ which matches
the coefficient of t n=n(cid:138) in Equation (1.18) with “a” replaced by (cid:11) and “b” replaced by (cid:12) where
(cid:11)
p5/=2. This means that the EGF for Fibonacci numbers is
p5/=2 and (cid:12)
(cid:11)n/=.(cid:12)
.1
.1
(cid:0)
(cid:0)
D
(cid:0)
D
C
(cid:16)e(cid:12) t
(cid:0)
e(cid:11)t (cid:17) =.(cid:12)
(cid:11)/
(cid:0)
D
F0
C
F1t=1(cid:138)
C
F2t 2=2(cid:138)
C
F3t
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2)
(cid:11)/
(cid:0)
D
F0
C
F1t=1(cid:138)
C
F2t 2=2(cid:138)
C
F3t 3=3(cid:138)
C (cid:1) (cid:1) (cid:1) C
Fnt n=n(cid:138)
C (cid:1) (cid:1) (cid:1)
;
(1.19)
where Fn is the nth Fibonacci number.
Another reason for the popularity of EGF is that they have interesting product and com-
position formulas. See Chapter 2 for details. The EGF has xn=n(cid:138) as multiplier. Some researchers
have extended it by replacing n(cid:138) by the nth Fibonacci number or its factorial Fn(cid:138) resulting in “Fi-
bonential” GF. Several extensions of EGF are available for specific problems. One of them is
1xn=n(cid:138), which has applications in al-
the tree-like GF (TGF) defined as F .x/
Pn
gorithm analysis, coding theory, etc.
0 annn
(cid:0)
D
(cid:21)
Example 1.7 (kth derivative of EGF) Prove that DkH.x/
P1n
D
D
0 an
C
kxn=n(cid:138)
a3x3=3(cid:138)
a2x2=2(cid:138)
akxk=k(cid:138)
n
C
D
C
C
a0
H.x/
a1x=1(cid:138)
C (cid:1) (cid:1) (cid:1) C
D
C (cid:1) (cid:1) (cid:1)
a2
C
1 for each term to get H 0.x/
Solution 1.7 Consider
.
C (cid:1) (cid:1) (cid:1)
Take the derivative w.r.t. x of both sides. As a0 is a constant, its derivative is zero. Use
xn
derivative of xn
(cid:0)
(cid:3)
C (cid:1) (cid:1) (cid:
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.t. x of both sides. As a0 is a constant, its derivative is zero. Use
xn
derivative of xn
(cid:0)
(cid:3)
C (cid:1) (cid:1) (cid:1) C
akxk
1=.k
. Differentiate again (this time a1 being a constant vanishes) to
1/(cid:138)
(cid:0)
(cid:0)
get H 00.x/
a3x=1(cid:138)
pro-
cess k times. All terms whose coefficients are below ak will vanish. What remains is
H .k/.x/
1x=1(cid:138)
. This can be expressed using the summation
C (cid:1) (cid:1) (cid:1)
notation introduced in Chapter 1 as H .k/.x/
k/(cid:138). Using the change of
indexvar introduced in Section 1.5 this can be written as H .k/.x/
kxn=n(cid:138). Now if
P1n
D
we put x
(cid:0)
D
0, all higher order terms vanish except the constant ak.
k anxn
P1n
(cid:0)
D
. Repeat
a3x2=2(cid:138)
a4x2=2(cid:138)
2x2=2(cid:138)
C (cid:1) (cid:1) (cid:1) C
a2x=1(cid:138)
akxk
k=.n
2=.k
C (cid:1) (cid:1) (cid:1)
0 an
this
2/(cid:138)
ak
ak
ak
a1
C
D
D
C
C
D
C
D
C
(cid:0)
C
C
C
(cid:0)
D
Example 1.8 (OGF of balls in urns) Find the OGF for the number of ways to distribute n dis-
tinguishable balls into m distinguishable urns (or boxes) in such a way that none of the urns is
empty.
Solution 1.8 Let un;m denote the total number of ways. Assume that i th urn has ni balls
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uishable balls into m distinguishable urns (or boxes) in such a way that none of the urns is
empty.
Solution 1.8 Let un;m denote the total number of ways. Assume that i th urn has ni balls so
n where each ni >
that all urn contents should add up to n. That gives n1
C (cid:1) (cid:1) (cid:1) C
x3=3(cid:138)
0. Let x=1(cid:138)
denote the GF for just one urn (where we have omitted
the constant term due to our assumption that the urn is not empty; meaning that there must
be at least one ball in it). As there are m such urns, the GF for all of them taken together is
/m. The coefficient of xn is what we are seeking. This is the sum
.x=1(cid:138)
x2=2(cid:138)
x2=2(cid:138)
x3=3(cid:138)
C (cid:1) (cid:1) (cid:1)
nm
n2
C
C
C
D
C
C
C (cid:1) (cid:1) (cid:1)
14
1. TYPES OF GENERATING FUNCTIONS
Pn1
n2
C
C(cid:1)(cid:1)(cid:1)C
nm
D
n n(cid:138)=(cid:140)n1(cid:138)n2(cid:138)
nm(cid:138)(cid:141). Denote the GF for un;m by H.x/. Then
(cid:1) (cid:1) (cid:1)
n
X
m
k
D
H.x/
D
akxk=k(cid:138)
.x=1(cid:138)
C
D
x2=2(cid:138)
x3=3(cid:138)
C
C (cid:1) (cid:1) (cid:1)
/m;
(1.20)
where ak
each urn gets one ball), and upper limit is n. The RHS is easily seen to be .ex
using binomial theorem to get
uk;m and the lower limit is obviously m
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)
/m;
(1.20)
where ak
each urn gets one ball), and upper limit is n. The RHS is easily seen to be .ex
using binomial theorem to get
uk;m and the lower limit is obviously m (because this corresponds to the case where
1/m. Expand it
D
(cid:0)
.ex
1/m
(cid:0)
D
emx
(cid:0)
!
m
1
e.m
(cid:0)
1/x
C
!
m
2
e.m
(cid:0)
2/x
1/m:
.
(cid:0)
(cid:0) (cid:1) (cid:1) (cid:1)
(1.21)
Expand each of the terms of the form ekx and collect coefficients of xn to get the answer as
un;m
mn
(cid:0)
D
!
m
1
.m
(cid:0)
1/n
C
!
m
2
.m
2/n
(cid:0)
1/m
(cid:0)
.
(cid:0)
(cid:0) (cid:1) (cid:1) (cid:1)
1 m
m
(cid:0)
!
1
(1.22)
because the last term in the above expansion does not have xn.
Example 1.9 (EGF of Laguerre polynomials) Prove that the Laguerre polynomials are the
EGF of .
0; 1; 2; : : : ; n.
1/k(cid:0)
n
k(cid:1) for k
(cid:0)
D
Solution 1.9 The Laguerre polynomials are defined as
Ln.x/
D
n
X
0
k
D
!
1/k n
k
.
(cid:0)
=k(cid:138)xk:
n
Writing this as P
k
polynomial.
0.
(cid:0)
D
n
1/k(cid:0)
k(cid:1).xk=k(cid:138)/ shows that the EGF of .
(1.23)
1/k(cid:0)
n
k(cid:1) is the classical Laguerre
(cid:0)
1.6
POCHHAMMER GENERATING FUNCTIONS
Analogous to the formal power series expansions given above, we could also define Poch
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)
n
k(cid:1) is the classical Laguerre
(cid:0)
1.6
POCHHAMMER GENERATING FUNCTIONS
Analogous to the formal power series expansions given above, we could also define Pochhammer
GF as follows. Here the series remains the same but the dummy variable is either rising factorial
or falling factorial type.
1.6.1 RISING POCHHAMMER GF (RPGF)
If (a0; a1; : : :) is a sequence of numbers, the RPGF is defined as
RPGF.x/
akx.k/;
D
X
0
k
(cid:21)
(1.24)
where x.k/
(cid:0)
tained where the known coefficients follow the Pochhammer factorial as
1/ : : : .x
1/. Note that this is different from the OGF or EGF ob-
x.x
D
C
C
k
1.6. POCHHAMMER GENERATING FUNCTIONS 15
RPGF.x/
a.k/xk;
n
X
0
k
(cid:21)
D
where a.k/
a.a
C
D
1/ : : : .a
k
C
(cid:0)
1/. An example of an EGF of this type is
b
x/(cid:0)
.1
(cid:0)
D
1
X
0
k
D
which is the EGF for Pochhammer numbers b.k/.
b.k/xk=k(cid:138)
1.6.2 FALLING POCHHAMMER GF (FPGF)
The dummy variable is of type falling factorial in this type of GF.
FPGF.x/
akx.k/:
D
X
0
k
(cid:21)
(1.25)
(1.26)
(1.27)
Consider Stirling number of second kind. The FPGF is xn because P
OGF of Stirling number of first kind is P
of first kind is P1n
D
k S.n; k/xn=n(cid:138)
ond kind is P1n
D
constants to get falling EGF for Pochhammer numbers b.k/.
xn. The
0 S.n; k/x.k/
x.n/. The
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; k/xn=n(cid:138)
ond kind is P1n
D
constants to get falling EGF for Pochhammer numbers b.k/.
xn. The
0 S.n; k/x.k/
x.n/. The EGF of Stirling number
x//k and that of Stirling number of sec-
1/k=k(cid:138). The falling factorial can also be applied to the
n
k
D
.1=k(cid:138)/.log.1
k s.n; k/xn=n(cid:138)
0 s.n; k/xk
D
.ex
D
C
D
D
(cid:0)
D
n
k
FPGF
D
1
X
0
k
b.k/xk=k(cid:138):
(1.28)
D
This shows that both the sequence terms and dummy variable can be written using Pochham-
mer symbol, which results in four different types of GFs for OGF and EGF. The falling
dummy variable technique was introduced by James Stirling in 1725 to represent an an-
0 akz.k/. Some ex-
alytic function f .z/ in terms of difference polynomials as f .z/
amples are z2
z.z
6z.z
3/
P1k
D
D
D
z; z4
C
7z.z
D
1/.z
z; z3
D
1/
3z.z
(cid:0)
2/
1/.z
1/.z
2/.z
(cid:0)
z, etc.
z.z
z.z
1/
1/
2/
C
C
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
C
(cid:0)
(cid:0)
C
(cid:0)
C
Example 1.10 Find OGF for the number of ways to color the vertices of a complete graph Kn.
Solution 1.10 Fix any of the nodes and give a color to it. There are n
there are n
1/.x
x.x
1 neighbors to it so that
1 choices. Continue arguing like this until a single node is left. Hence, the GF is
2/ : : : .x
1/ or x.n/ as the required
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/.x
x.x
1 neighbors to it so that
1 choices. Continue arguing like this until a single node is left. Hence, the GF is
2/ : : : .x
1/ or x.n/ as the required GF.
(cid:0)
n
(cid:0)
(cid:0)
(cid:0)
(cid:0)
C
16
1. TYPES OF GENERATING FUNCTIONS
1.7 OTHER GENERATING FUNCTIONS
There are many other GFs used in various fields. Some of them are simple modifications of the
dummy variable while others are combinations. A few of them are given below.
1.7.1 AUTO-COVARIANCE GENERATING FUNCTION
Consider the GF
P .z/
2 a0
a1.z
C
C
D
1=z/
C
a2.z2
C
1=z2/
C (cid:1) (cid:1) (cid:1) C
ak.zk
1=zk/
;
C (cid:1) (cid:1) (cid:1)
C
(1.29)
where ak’s are the known coefficients. This is used in some of the time series modeling. Separate
the z and 1=z terms to get a0
which can
C
C
be written as two separate GF as G1.z/
G2.1=z/. See Shishebor et al. [2006] for autocovari-
ance GFs for periodically correlated autoregressive processes.
C (cid:1) (cid:1) (cid:1) C
C (cid:1) (cid:1) (cid:1) C
and a0
a2=z2
a2z2
a1=z
a1z
C
C
C
Example 1.11 (OGF of Auto-correlation) If the auto-correlation sequence of a discrete time
process X(cid:140)n(cid:141) is given by R(cid:140)n(cid:141)
< 1, find the auto-correlation GF.
n
aj
j where
D
a
j
j
Solution 1.11 Split
n
as P(cid:0)
n
1=.1
a/
nt (cid:0)
a(cid:
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GF.
n
aj
j where
D
a
j
j
Solution 1.11 Split
n
as P(cid:0)
n
1=.1
a/
nt (cid:0)
a(cid:0)
at/.
1
D(cid:0)1
(cid:0)
C
C
the range from (
P1n
D
(cid:0)1
0 ant n. This simplifies to .a=t/.1
(cid:0)
to
1) and (0 to
a=t/(cid:0)
(cid:0)
)
1
1
C
to get
1=.1
at/
the OGF
a=.t
(cid:0)
D
(cid:0)
1.7.2
INFORMATION GENERATING FUNCTION (IGF)
n
Let p1; p2; : : : ; pn be the probabilities of a discrete random variable, so that P
1. The
k
D
IGF of a discrete random variable is defined as I.t/
k where auxiliary variable t is
real or complex depending on the nature of the probability distribution. Differentiate w.r.t. t
and put t
1 to get
n
P
k
1 pt
1 pk
D (cid:0)
D
D
D
H.p/
D (cid:0)
@[email protected]/
1
t
j
D
D (cid:0)
pk log.pk/
(1.30)
n
X
1
k
D
which is Shannon’s entropy. Differentiate m times and put t
1 to get
D
H.p/.m/
.@=@t/mI.t/
1
t
j
D
D
.
1/m
(cid:0)
1
(cid:0)
D
n
X
1
k
D
log.pk//m
pk.
(cid:0)
(1.31)
which is the expected value of mth power of
log.p/.
(cid:0)
1.7.3 GENERATING FUNCTIONS IN GRAPH THEORY
Graph theory is a branch of discrete mathematics that deals with relationships (adjacency and
incidence) among a set of finite number of elements. As a graph with n vertices is a subset of
1.7. OTHER GENERATING FUNCTIONS 17
n
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mathematics that deals with relationships (adjacency and
incidence) among a set of finite number of elements. As a graph with n vertices is a subset of
1.7. OTHER GENERATING FUNCTIONS 17
n
the set of (cid:0)
2(cid:1) pairs of vertices there exist a total number of 2.n
2/ graphs on n vertices. Hence, the
2/
.n
0 Gn;kxk where Gn;k is the number of graphs
GF for graphs with n vertices is Gn.x/
P
k
D
with n vertices and k edges. Total number of simple labeled graphs with n vertices and m edges
0 2.n
eC.x/ where C.x/ is the EGF of
is (cid:0)
connected graphs. These are further discussed in Chapter 4 (pp. 73).
.n
2/
m (cid:1). This in turn results in the EGF P1n
D
2/xn=n(cid:138)
D
D
1.7.4 GENERATING FUNCTIONS IN NUMBER THEORY
There are many useful GFs used in number theory. Some of them are discussed in subse-
quent chapters. Examples are Fibonacci and Lucas numbers, Bell numbers, Catalan num-
bers, Lah numbers, Bernoulli numbers, Stirling numbers, Euler’s (cid:27).n/, etc. The Lah num-
k
bers satisfy the recurrence L.n; k/
1; k/. It has EGF
.n
1; k
D
t//k. Replacing t by
t, this could also be written as
.1=k(cid:138)/.t=.1
P1n
D
1/k.1=k(cid:138)/.t=.1
1/nL.n; k/t n=n(cid:138).
.
0.
0 L.n; k/t n=n(cid:138)
D
t//k
1/L.n
C
(cid:0)
L.n
1/
C
(cid:0)
(cid:0)
(cid:0)
(cid:0)
C
P1n
D
D
(cid:0)
(cid:0)
(cid:0)
1.7.5 ROOK POLYNOMIAL GENERATING FUNCTION
(cid:2)
Consider an n
n chessboard. A rook can be placed in any cell such
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(cid:0)
(cid:0)
(cid:0)
1.7.5 ROOK POLYNOMIAL GENERATING FUNCTION
(cid:2)
Consider an n
n chessboard. A rook can be placed in any cell such that no two rooks appear in
any row or column. Let rk denote the number of ways to place k rooks on a chessboard. Obvi-
n(cid:138). The rook polynomial
ously r0
GF can now be defined as
n2 (because it can be placed anywhere), and rn
1 and r1
D
D
D
R.t/
r0
C
D
r1t
C
r2t 2
:
C (cid:1) (cid:1) (cid:1)
(1.32)
(cid:2)
D
n2
The rook polynomial for an n
n chessboard is given by P
ways to place one rook. Suppose there are k distinguishable rooks. The first can be place in
n2 ways, which marks-off one row and one column (because rooks attack horizontally
r1
1/2 squares remaining so that the second one can be placed in
or vertically). There are .n
1/2 ways and so on. As the rooks can be arranged among themselves in k(cid:138) ways, we get
.n
n
1/2=k(cid:138)t k. Multiply and divide by k(cid:138) and use
0 n2.n
the GF as P
k
D
2/ : : : .n
n.n
2/2 : : : .n
C
n
k(cid:1) to get the result.
(cid:0)
t k. There are r1
(cid:0)
1/=k(cid:138)
0 k(cid:138)(cid:0)
1/.n
D
(cid:0)
(cid:0)
(cid:0)
k
D
n
k
2
n
k(cid:1)
1/2.n
(cid:0)
k
(cid:0)
C
(cid:0)
(cid:0)
D
1.7.6 STIRLING NUMBERS OF SECOND KIND
There are two types of Stirling numbers—the first kind denoted by s.n
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k
(cid:0)
C
(cid:0)
(cid:0)
D
1.7.6 STIRLING NUMBERS OF SECOND KIND
There are two types of Stirling numbers—the first kind denoted by s.n; k/ and second kind
denoted by S.n; k/. Consider a set S with n elements. We wish to partition it into k.< n/
nonempty subsets. The S.n; k/ denotes the number of partitions of a finite set of size n into k
subsets. It can also be considered as the number of ways to distribute n distinguishable balls into
k indistinguishable cells with no cell empty.
18
1. TYPES OF GENERATING FUNCTIONS
The Stirling number of second kind satisfies the recurrence relation S.n; k/
1; k/
S.n
1; k
(cid:0)
(cid:0)
C
1/. The OGF for the Stirling numbers of the second kind is given by
kS.n
(cid:0)
D
S.n; k/xn
xk=(cid:140).1
D
x/.1
(cid:0)
(cid:0)
2x/.1
(cid:0)
3x/ : : : .1
kx/(cid:141):
(cid:0)
(1.33)
1
X
0
n
D
A simplified form results when n is represented as a function of k (displaced by another integer):
S.m
C
n; m/xn
1=(cid:140).1
x/.1
(cid:0)
(cid:0)
D
2x/.1
(cid:0)
3x/ : : : .1
mx/(cid:141):
(cid:0)
(1.34)
1
X
0
n
D
1.8
SUMMARY
This chapter introduced the basic concepts in GFs. Topics covered include structure of GFs,
types of GFs, ordinary and exponential GFs, etc. The Pochhammer GF for rising and falling
factorials are also introduced. Some special GFs like auto-covariance GFs, information GFs,
and those encountered in number theory and graph theory are briefly described. Multiplying
x/ results in the OGF of the partial sums of coefficients. This
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like auto-covariance GFs, information GFs,
and those encountered in number theory and graph theory are briefly described. Multiplying
x/ results in the OGF of the partial sums of coefficients. This result is used
an OGF by 1=.1
extensively in Chapter 4.
(cid:0)
C H A P T E R 2
19
Operations on Generating
Functions
After finishing the chapter, readers will be able to
(cid:1) (cid:1) (cid:1)
• do arithmetic on generating functions.
• find linear combination of generating functions.
• differentiate and integrate generating functions.
• apply left-shift and right-shift of generating functions.
• find convolution and powers of generating functions.
2.1 BASIC OPERATIONS
D
2n
The first chapter explored several simple sequences. As a sequence is indexed by natural numbers,
the best way to express an arbitrary term of a sequence seems to be a closed form as a function
1 expresses the nth term as a function of n. Some of
of the index .n/. For example, an
the series encountered in practice cannot be described succinctly as shown above. A recurrence
relation connecting successive terms may give us good insights for some sequences. Such a re-
currence suffices in most of the problems, especially those used in updating the terms. But it
may be impractical to use a recurrence to calculate the coefficients for very large indices (say 10
millionth term). Given a recurrence for two sequences an and bn, how do we get a recurrence
for an
an?. This is where the merit of introducing a GF becomes apparent. It
allows us to derive new GFs from existing or already known ones.
bn or for c
C
(cid:0)
(cid:3)
0 pkt k, or
An infinite sum is denoted by either explicit enumeration of the range as P1k
D
0 pkt k where the upper limit is to be interpreted appropriately. Most of the following
as Pk
(cid:21)
results are apparent when the coefficient of the power series are represented as sets. For ex-
ample, if (a0; a1; : : :) and (b0; b1; : : :) are, respectively, the coefficients of A
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fficient of the power series are represented as sets. For ex-
ample, if (a0; a1; : : :) and (b0; b1; : : :) are, respectively, the coefficients of A.t/ and B.t /, then
(a0
B.t/; while (c0; c1; : : :) are the coefficients of A.t/
etc., cn
(cid:0)
a1b0,
anb0. Two GFs, say F .t/ and G.t/ are exactly identical when
b1; : : :) are the coefficients of the sum and difference A.t/
B.t/, where c0
B.t/ and A.t/
C
a0b0, c1
b0; a1
a1bn
a0bn
a0b1
(cid:6)
(cid:6)
C
D
D
(cid:3)
1
D
C
(cid:0)
C (cid:1) (cid:1) (cid:1) C
20
2. OPERATIONS ON GENERATING FUNCTIONS
each of the coefficients of t k are equal for all k. This holds good for OGF, EGF, and other
aforementioned GFs.
Let F .t/ and G.t/ defined as
F .t/
G.t/
a0
b0
D
D
C
C
a1t
b1t
C
C
a2t 2
b2t 2
C (cid:1) (cid:1) (cid:1) D
C (cid:1) (cid:1) (cid:1) D
0 akt k
P1k
D
0 bkt k
P1k
D
(2.1)
(2.2)
be two arbitrary OGF and with compatible coefficients. Here, ak (respectively bk) is the coef-
ficient of t k in the power series expansion of F .t/ (resp. G.t/).
2.1.1 EXTRACTING COEFFICIENTS
Quite often, the GF approach allows us to find a compact representation for a sequence at hand,
and to extract the terms of a sequence from its GF using simple techniques (power series expan
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.1 EXTRACTING COEFFICIENTS
Quite often, the GF approach allows us to find a compact representation for a sequence at hand,
and to extract the terms of a sequence from its GF using simple techniques (power series expan-
sion, differentiation, etc.). Thus it is a two-way tool. Coefficients can often be extracted easily
from a GF. But there are situations where we may have to use other mathematical techniques
like partial fractions, differentiation, logarithmic transformations, synthetic division etc. Sup-
pose a GF is in the form p.x/=q.x/ where p.x/ is either a polynomial or a constant. If the degree
of p.x/ is less than that of q.x/, and q.x/ contains products of functions, we may split the entire
expression using partial fractions to simplify the coefficient extraction. If the degree of p.x/ is
greater than or equal to that of q.x/, we may use synthetic division to get an expression of the
p1.x/=q.x/, where the degree of p1.x/ is less than that of q.x/, and proceed with
form r.x/
partial fraction decomposition of p1.x/=q.x/. These are illustrated in examples below.
C
2.1.2 ADDITION AND SUBTRACTION
The sum and difference of F .t/ and G.t/ are defined as
F .t/
G.t/
.a0
b0/
.a1
C
(cid:6)
(cid:6)
D
b1/t
C
.a2
(cid:6)
(cid:6)
b2/t 2
C (cid:1) (cid:1) (cid:1) D
.ak
(cid:6)
bk/t k:
(2.3)
1
X
0
k
D
Thus, the above result can be extended to any number of GFs as F .t/
,
(cid:6) (cid:1) (cid:1) (cid:1)
(cid:6)
where each of them are either OGF or EGF. Hence, F .t/
H.t/ is a valid OGF if
each of them is an OGF. This is a special case of the linear combination discussed below. Some
authors denote this compactly as .F
H /.t/. This has applications in several field
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is a valid OGF if
each of them is an OGF. This is a special case of the linear combination discussed below. Some
authors denote this compactly as .F
H /.t/. This has applications in several fields like
statistics where we work with sums of independently and identically distributed (IID) random
variables.
H.t/
G.t/
G.t/
C
(cid:6)
C
G
(cid:0)
(cid:0)
2.1.3 MULTIPLICATION BY NON-ZERO CONSTANT
This is called a scalar product or scaling, and is defined as cF .t/
akt k
(cid:3)
c ak, and c is a non-zero constant. This technique allows us
c P1k
D
0 bkt k where bk
P1k
D
0 c
D
D
0 akt k
P1k
D
D
D
to represent a variety of sequences using a constant times a single GF. For example, if there
are multiple geometric series that differ only in the common ratio, we could represent all of
them using this technique. As constant c is arbitrary, we could also divide a GF (OGF, EGF,
Pochhammer GF, etc.) by a non-zero constant.
2.1. BASIC OPERATIONS 21
2.1.4 LINEAR COMBINATION
(cid:3)
C
Scalar multiplication can be extended to a linear combination of OGF as follows. Let c and
dbk/t k (where the missing
d be two non-zero constants. Then cF .t/
dG.t/
D
). The constants can be positive or negative, which results in a variety of new GFs.
operator is
This also can be extended to any number (n
2) of GFs. As a special case, the equality of two
GFs mentioned in the first paragraph can be proved as follows. Suppose F .t/ and G.t/ are two
1 and d
GFs. Assume for simplicity that both are OGF. Then by taking c
1, we have
D (cid:0)
bk/t k. Equating to zero
0 bkt k, or equivalently P1k
F .t/
P1k
D
D
shows that this is true only when each of the a0ks and b0ks are equal, proving the
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ating to zero
0 bkt k, or equivalently P1k
F .t/
P1k
D
D
shows that this is true only when each of the a0ks and b0ks are equal, proving the result. The same
argument holds for EGF, Pochhammer GF, and other GFs.
0 akt k
P1k
D
D C
(cid:0)
P1k
D
0.cak
0.ak
G.t/
D
C
(cid:21)
(cid:0)
(cid:0)
Example 2.1 (OGF of Arithmetic Progression) Find the OGF of a sequence (a; a
2d; : : : ; a
ference d .
C
1/d; : : :) in arithmetic progression (AP) with initial term ‘a’ and common dif-
d; a
.n
C
C
(cid:0)
Solution 2.1 Write the OGF as
F .t/
.a
a
C
C
D
d /t
C
.a
C
2d /t 2
C (cid:1) (cid:1) (cid:1) C
nd /t n
.a
C
:
C (cid:1) (cid:1) (cid:1)
This can be split into two separate OGFs as
F .t/
a (cid:2)1
t
C
C
D
t 2
C (cid:1) (cid:1) (cid:1)
This gives F .t/
a=.1
t/
(cid:0)
C
D
dt=.1
(cid:0)
(cid:3)
C
t/2.
2.1.5 SHIFTING
dt (cid:2)1
2t
C
C
3t 2
C (cid:1) (cid:1) (cid:1) C
nt n
(cid:0)
1
(cid:3) :
C (cid:1) (cid:1) (cid:1)
(2.4)
(2.5)
If we have a sequence (a0; a1; : : :), shifting it right by one position means to move everything
to the right by one unit and fill the vacant slot on the left by a zero. This results in the se-
quence
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sequence (a0; a1; : : :), shifting it right by one position means to move everything
to the right by one unit and fill the vacant slot on the left by a zero. This results in the se-
quence (0; a0; a1; : : :), which corresponds to b0
1. Similarly, shifting
left means to move everything to the left by one position and discarding the very first term
0. For finite sequence, we may have to
1 for n
on the left, which corresponds to bn
fill the vacant terms on the right-most position by zeros. Let F .t/
a0
be the OGF for (a0; a1; : : :). Then tF .t/ is an OGF for (0; a0; a1; : : :). In general t mF .t/
a0t m
C (cid:1) (cid:1) (cid:1)
D
0; 0; : : : 0, a0; a1; a2; : : :) where the “zero
. This is the OGF of ((cid:130) (cid:133)(cid:132) (cid:131)
1 for n
a1t m
a2t m
a2t 2
0; bn
a1t
an
an
D
D
D
D
C
C
(cid:21)
(cid:21)
C
(cid:0)
1
2
C
C
C
C
C (cid:1) (cid:1) (cid:1)
22
2. OPERATIONS ON GENERATING FUNCTIONS
block” in the beginning has m zeros. Simple algebraic operations can be performed to show that
(cid:0)F .t/
a0
(cid:0)
(cid:0)
a1t
(cid:0)
a2t 2
(cid:0) (cid:1) (cid:1) (cid:1) (cid:0)
am
(cid:0)
1t m
1(cid:1) =t m
(cid:0)
am
am
1t
C
C
C
am
C
D
2t 2
C (cid:1) (cid:1)
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m
1(cid:1) =t m
(cid:0)
am
am
1t
C
C
C
am
C
D
2t 2
C (cid:1) (cid:1) (cid:1)
(2.6)
which is a shift-left by m positions. Shifting is applicable for both OGF and EGF, and will be
used in the following paragraphs.
Example 2.2 Let G.t/ be the OGF of a sequence (a0; a1; : : :). What is the sequence whose
OGF is .1
t/G.t/?
C
Solution 2.2 Write .1
a1; a1
(a0; a0
C
C
C
t/G.t/
G.t/
tG.t/. It follows easily that this is the OGF of
a2; : : :) because tG.t/ is the aforementioned right-shifted sequence.
D
C
Example 2.3 Find the OGF of the sequence .1; 3; 7; 15; 31; : : :).
Solution 2.3 The kth term is obviously (2k
1.2k
the given sequence. Then F .t/
P1k
D
1
2t/(cid:0)
.1
is P1k
(cid:0)
D
D
t/
t=.1
t/. Hence, F .t/
(cid:0)
1.
2t/(cid:141)
D
C (cid:1) (cid:1) (cid:1) D
2t /
(cid:0)
1 2kt k
C
1=.1
4t 2
D
2t
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
1. Let F .t/ denote the OGF of
1) for k
1/t k. Split this into two series. The first one
1. The second one is
(cid:21)
1 t k
P1k
D
2t 2/=(cid:140).1
(cid:0)
2t
t=.1
t/.1
(cid:0)
(cid:0)
D (cid:0)
(cid:0)
(cid:0)
C
1, which
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cid:140).1
(cid:0)
2t
t=.1
t/.1
(cid:0)
(cid:0)
D (cid:0)
(cid:0)
(cid:0)
C
1, which simplifies to .1
(cid:0)
Example 2.4 Find the OGF and EGF of the sequence .12; 32; 52; 72; : : :/.
(cid:21)
Solution 2.4 Let F .t/ denote the OGF of the given sequence. The kth term is obviously .2k
1/2 for n
0. Hence, F .t/
D
0 kt k
4 P1k
into three terms to get F .t/
D
D
been encountered in previous chapter. Thus, we get F .t/
1=.1
C
1/2
1 and split
4k
0 t k. All of these have already
C
P1k
D
4t.1
C
1/2t k. Expand .2k
t/.
(cid:0)
Next let H.t/ denote the EGF of the given sequence. Then H.t/
P1k
D
4 P1k
D
0.2k
C
0 k2t k
1/2t k=k(cid:138).
C
D
t/=.1
4=.1
4k2
t/2
t/3
C
C
C
C
D
C
(cid:0)
(cid:0)
1/2
Expand .2k
C
0 t k=k(cid:138). Write k2
4 P1k
D
EGF’s. Then combine the first and second series. This results in
D
1 and split into three terms to get H.t/
1/
C
0 kt k=k(cid:138)
D
P1k
D
4k2
k.k
4k
D
C
C
C
(cid:0)
0.2k
P1k
D
4 P1k
D
C
0 k2t k=k(cid:138)
C
k and split the first term into two
D
H.t/
D
4t 2 1
X
2
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P1k
D
C
0 k2t k=k(cid:138)
C
k and split the first term into two
D
H.t/
D
4t 2 1
X
2
k
D
t k
2=.k
(cid:0)
2/(cid:138)
(cid:0)
C
8t
t k
(cid:0)
1=.k
1/(cid:138)
(cid:0)
C
1
X
1
k
D
1
X
0
k
D
t k=k(cid:138):
(2.7)
This simplifies to H.t/
.4t 2
8t
C
C
1/et .
D
2.1.6 FUNCTIONS OF DUMMY VARIABLE
The dummy variable in a GF can be assumed to be continuous in an interval. Any type of
transformation can then be applied on them within that region. A simple case is the scaling of
t as ct in an OGF
2.1. BASIC OPERATIONS 23
ak.ct/k
F .ct/
D
X
0
k
(cid:21)
D
X
0
k
(cid:21)
.ckak/t k;
(2.8)
which is an OGF of
taking c
ckak
f
1=2, we get the series
g
D
ak=2k
f
.
g
where c can be any real number or fractions of the form p=q. By
Example 2.5 If F .t/
a12t 12
a8t 8
a1t
a0
/ and (ii) .a0
C
D
C
C (cid:1) (cid:1) (cid:1)
a2t 2
a2t 2
C
(cid:0)
C (cid:1) (cid:1) (cid:1)
a4t 4
C
is an OGF, express the OGFs: (i) .a0
a8t 8
(cid:0)
C (cid:1) (cid:1) (cid:1)
/ in terms of F .t/.
a4t 4
C
C
Solution 2.5
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a8t 8
(cid:0)
C (cid:1) (cid:1) (cid:1)
/ in terms of F .t/.
a4t 4
C
C
Solution 2.5 As the first sequence contains powers of t 4, this has OGFF .t 4/. The second se-
quence has alternating signs. As the imaginary constant i
1, the second sequence can be
generated using F ..it/2/ because i 2
p
1, and so on.
1; i 4
D
(cid:0)
D (cid:0)
D C
Example 2.6 If F .t/
a16t 16
a8t 8
a4t 4
C
C
a1t
a2t 2
a3t 3
C
/ in terms of F .t/.
C
C
a0
D
C (cid:1) (cid:1) (cid:1)
C (cid:1) (cid:1) (cid:1)
express the OGF .a0
a1t
C
C
a2t 2
C
Solution 2.6 Writing t 4
F .t 2n
D
0; 1; 2; 3; : : :.
/ for n
D
.t 2/2, t 8
D
.t 2/3, etc., it is easy to see that the OGF of RHS is
Example 2.7 Find the OGF of the sequence whose nth term is ..n
1/=2n/ for n
0.
(cid:21)
C
Solution 2.7 Let F .t/ denote the OGF
F .t/
..n
C
D
X
0
n
(cid:21)
1/=2n/ t n
1/.t=2/n:
D
.n
X
0
n
(cid:21)
C
(2.9)
(cid:0)
(cid:0)
(cid:0)
s/(cid:0)
t=2/2.
Putting s
is .1
2. Thus, the OGF of original sequence
t=2 shows that this is the OGF of .1
2 or 1=.1
D
t=
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)
t=2/2.
Putting s
is .1
2. Thus, the OGF of original sequence
t=2 shows that this is the OGF of .1
2 or 1=.1
D
t=2/(cid:0)
Another application of the change of dummy variables is in evaluating sums involv-
n
1
1(cid:1)ak. This sum is the coefficient of
ing binomial coefficients. Consider the sum P
k
t n in the power series expansion in which the dummy variable is changed as t=.1
t/.
n
n
t//, where F .t/ is the OGF of ak. Similarly,
Symbolically, P
1 (cid:0)
k
k
D
n
1
k(cid:0)
1(cid:1)ak
P
(cid:0)
(cid:0)
k
(cid:0)
1
1(cid:1)ak
D
(cid:140)t n(cid:141)F .t=.1
(cid:140)t n(cid:141)F .t=.1
t//.
(cid:0)
(cid:0)
D
n
1 (cid:0)
k
1/n
1.
(cid:0)
(cid:0)
C
(cid:0)
(cid:0)
(cid:0)
n
k
D
D
24
2. OPERATIONS ON GENERATING FUNCTIONS
2.1.7 CONVOLUTIONS AND POWERS
The literal meaning of convolution is “a thing, a form or shape that is folded in curved twist or
tortuous windings that is complex or difficult to follow.” A convolution in GFs is defined for
two sequences (usually different or time-lagged as in DSP) with the same number of elements
as the sum of the products of terms taken in opposite directions. As an example, the convolution
of .1; 2; 3/ and .4; 5; 6/ is 1
28. Consider the product of two GFs
2
6
3
5
4
(cid:3)
C
(cid:3)
C
(cid:3)
D
k
X
0
j
D
1
X
0
k
akt k
(cid:3)
1
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3
5
4
(cid:3)
C
(cid:3)
C
(cid:3)
D
k
X
0
j
D
1
X
0
k
akt k
(cid:3)
1
X
0
k
bkt k
D
1
X
0
k
ckt k where ck
D
aj bk
j :
(cid:0)
(2.10)
D
D
D
k aj bi . As the order of
This is called the convolution, which can also be written as ck
k
j bj . The coefficients of a convolution
the OGF can be exchanged, it is the same as P
j
all lie on a diagonal which is perpendicular to the main diagonal (Table 2.1). This shows that a
convolution of two sequences with compatible coefficients can be represented as the product of
their GFs. This has several applications in selection problems, signal processing, and statistics.
This can be expressed as
0 ak
Pi
D
D
C
D
(cid:0)
j
F .t/G.t/
D
X
0
k
(cid:21)
.a0bk
a1bk
C
1
(cid:0)
C (cid:1) (cid:1) (cid:1) C
akb0/ t k:
(2.11)
Suppose there are two disjoint sets A and B. If the GF for selecting items from A is A.t/ and
from B is B.t/, the GF for selecting from the union of A and B.A
B/ is A.t/B.t/. This can
be applied any number of times. Thus, if A.t/; B.t/, and C.t/ are three OGFs with compatible
coefficients, the OGF of A.t/B.t/C.t/ has mth coefficient um
[
m ai bj ck.
Pi
j
k
D
C
C
D
Table 2.1: Convolution as diagonal sum
Example 2.8 Find the OGF of the product of .1; 2; 3; 4; : : :/ and .1; 2; 4; 8; 16; : : :/.
Solution 2.8 The first OGF is obviously
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1; 2; 3; 4; : : :/ and .1; 2; 4; 8; 16; : : :/.
Solution 2.8 The first OGF is obviously 1=.1
volution is the product of the OGF 1=(cid:140).1
(cid:0)
t/2.1
(cid:0)
t/2 and second one is 1=.1
2t/. The con-
2t/(cid:141). Use partial fractions to break this as
(cid:0)
(cid:0)
b0b1xb2x2b3x3b4x4…a0a0b0a0b1xa0b2x2a0b3x3a0b4x4…a1xa1b0xa1b1x2a1b2x3a1b3x4a1b4x5…a2x2a2b0x2a2b1x3a2b2x4a2b3x5a2b4x6…a3x3a3b0x3a3b1x4a3b2x5a3b3x6a3b4x7…………………a4x4 t/2
(cid:0)
(cid:0)
(cid:0)
t/
C
B=.1
C =.1
2t /. Take .1
A=.1
C
and equate numerators on LHS and RHS to get A.1
C
Now equate constant, coefficient of t, and t 2 to get 3 equations A
2C
A
t /
D (cid:0)
D
t/2 with RoC 0 < t < 1=2.
0; 2A
C
2, from which C
4 and B
(cid:0)
t /.1
(cid:0)
2t /
1=.1
t/.1
C
D
(cid:0)
(cid:0)
(cid:0)
t/2.1
D
D (cid:0)
(cid:0)
(cid:0)
C
0. Multiply the first equation by 2 and subtract from the second one to get
C
C
C
1. Thus, the OGF
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D (cid:0)
(cid:0)
(cid:0)
C
0. Multiply the first equation by 2 and subtract from the second one to get
C
C
C
1. Thus, the OGF of the product is 4=.1
2t /
(cid:0)
(cid:0)
2=.1
(cid:0)
2.1. BASIC OPERATIONS 25
2t / as a common denominator
1.
B.1
t /2
2t /
C
(cid:0)
B
C.1
(cid:0)
1; 3A
C
D
D
2B
Now consider two EGFs H1.t /
P1k
D
D
0 akt k=k(cid:138) and H2.t/
P1k
D
D
0 bkt k=k(cid:138). Their
product H1.t/H2.t / has “exponential convolution”
akt k=k(cid:138)
1
X
0
k
D
(cid:3)
1
X
0
k
D
bkt k=k(cid:138)
D
ckt k=k(cid:138)
aj bk
(cid:0)
j k(cid:138)=.j (cid:138).k
j /(cid:138)/
(cid:0)
D
aj bk
j :
(cid:0)
1
X
0
k
!
D
k
j
k
X
0
j
D
ck
D
k
X
0
j
D
(2.12)
(2.13)
As the order of the EGF can be exchanged, it is the same as ck
k
j (cid:1)ak
0 (cid:0)
As convolution is commutative and associative, we could write F .t /G.t /
k
P
j
D
D
D
F .t /G.t /H.t /
G.t/H.t /F .t /
tion and subtraction. This means that F .t /(cid:140)G.t /
an OGF are repeated convolutions. Thus, if F .t /
k
P
P1k
j
D
0 ckt k where ck
0 aj ak
D
D
(cid:
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40)G.t /
an OGF are repeated convolutions. Thus, if F .t /
k
P
P1k
j
D
0 ckt k where ck
0 aj ak
D
D
(cid:0)
G.t /F .t / and
H.t/F .t /G.t /, etc. Similarly, product distributes over addi-
F .t /G.t /
F .t /H.t /. Powers of
D
0 akt k then F .t /
F .t /2
H.t /(cid:141)
F .t /
(cid:6)
(cid:6)
D
j , which can also be expressed as
P1k
D
(cid:3)
D
D
j bj .
(cid:0)
D
D
(cid:140)F .t /(cid:141)2
.a0ak
a1ak
C
1
(cid:0)
C (cid:1) (cid:1) (cid:1) C
aka0/ t k:
D
X
0
k
(cid:21)
(2.14)
As an illustration of powers of OGF, consider the number of binary trees Tn with n nodes where
one unique node is designated as the root. Assume that there are k nodes on the left subtree and
n
n/. It can be shown that it satisfies the recurrence
k nodes on the right subtree (so that the total number of nodes is k
k/
.n
C
D
C
(cid:0)
(cid:0)
(cid:0)
(cid:0)
1
1
1
Tn
D
n
1
(cid:0)
X
0
k
D
TkTn
k
1
(cid:0)
(cid:0)
for n
1
(cid:21)
(2.15)
because k can take any value from 0 (right skewed tree) to n
1 (left skewed tree). The form
of the recurrence suggests that it is of the form of some power (Table 2.2). But as the indexvar
0 Tnxn.
is varying from 0 to n
Multiply both sides of (2.15) by xn and sum over n
1, it is not an exact power as shown below. Let T .x/
P1n
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0 Tnxn.
is varying from 0 to n
Multiply both sides of (2.15) by xn and sum over n
1, it is not an exact power as shown below. Let T .x/
P1n
D
to get
1 to
D
(cid:0)
(cid:0)
D
1
Tnxn
D
1
X
1
n
D
1
X
1
n
D
n
1
(cid:0)
X
0
k
D
TkTn
1
(cid:0)
(cid:0)
kxn for n
1;
(cid:21)
(2.16)
26
2. OPERATIONS ON GENERATING FUNCTIONS
Table 2.2: Summary of convolutions and powers
T0. Assume that T0
where we have used x instead of t as dummy variable for convenience. As the LHS indexvar is
1. Take one x outside the summation
varying from 1 onward, it is T .x/
on the RHS and write xn
k. Then the RHS is easily seen to be x.T .x//2, so that
.1
we get T .x/
(cid:6)
p1
4x/=.2x/
as the required OGF for the number of binary trees with n nodes. See Chapter 4 for another
application of convolutions to matched parentheses.
x.T .x//2. Solve this as a quadratic equation in T .x/. This gives T .x/
1
4x/=.2x/. As T .x/
0, take negative sign to get T .x/
(cid:0)
xkxn
! 1
as x
p1
!
.1
D
D
D
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
1
1
Example 2.9 Find the OGF of Catalan numbers that satisfy the recurrence C0
C0Cn
D
1C0 for n > 1. Prove that Catalan numbers are always integers.
C1Cn
1; Cn
Cn
D
2
1
(cid:0)
C
(cid:0)
C (cid:1) (cid:1) (cid:1) C
(cid:
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.
C1Cn
1; Cn
Cn
D
2
1
(cid:0)
C
(cid:0)
C (cid:1) (cid:1) (cid:1) C
(cid:0)
Solution 2.9 Let F .t / denote the OGF of Catalan numbers. Write the convolution as a sum
Cn
j . As per Table 2.2 entry row 2, this is the coefficient of a power. As done
1
(cid:0)
above, multiply by t n and sum over 0–
1
0 Cj Cn
to get
n
P
j
(cid:0)
D
D
(cid:0)
1
1
n
(cid:0)
X
0
j
1
X
0
n
1
X
0
n
Cnt n
D
Cj Cn
1
(cid:0)
(cid:0)
j t n for n
1:
(cid:21)
(2.17)
j
1
(cid:0)
(cid:0)
D
D
D
t j t n
1, we get F .t /
4t /=.2t /. As t
D
t. Then the RHS is easily found to be the coefficients of t F .t /2. As
t F .t /2
0; .1
Write t n
C0
(cid:6)
p1
(as it is 1/0 form), but the other root is
0/0 form, which by L’Hospitals rule tends to 1. Alternatively, the plus sign results in negative
coefficients but as Catalan numbers are always positive integers, we have to take the minus sign.
1. Solve this as a quadratic in F .t / to get F .t /
(cid:3)
D
!
4t /=.2t /
! 1
C
p1
D
(cid:0)
.1
C
D
(cid:0)
TypeExpressionCoeffi cient (ck)OGF Product∞ aiti ∗∞ bjtj k ajbk–jOGF Power∞ aiti2k ajak–jOGF Product (of 3)∞ ahth ∗∞ biti ∗∞ cjtj h+i+j=k, k
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bjtj k ajbk–jOGF Power∞ aiti2k ajak–jOGF Product (of 3)∞ ahth ∗∞ biti ∗∞ cjtj h+i+j=k, k≥0 ahbicj EGF Product∞ aiti /i! ∗∞ bjtj /j!k kajbk–jEGF Power∞ aiti /i!2k kajak–jEGF Product (of 3)∞ ah th ∗∞ bi ti ∗∞ cj tjh+i+j=k ahbicj k! i=0i=0h=0h=0i=0i=0i=0i=0j=0j=0j=0j=0j=0j=0j=0 jj=0 jh!i!j!h!i!j! Hence, we take F .t/
p1
t using
C
.1
(cid:0)
D
p1
(cid:0)
4t/=.2t/, which is the same as in previous example. Expand
2.1. BASIC OPERATIONS 27
p1
t
C
D
.1
C
t/1=2
D
1
X
0
k
D
!
1=2
k
t k
1
C
D
!
1=2
k
t k
1
X
1
k
D
because (cid:0)
1=2
0 (cid:1)
D
1. Now use (cid:0)
1=2
k (cid:1)
21
2k.
(cid:0)
1/k
(cid:0)
(cid:0)
D
1(cid:0)
2k
k
2
1 (cid:1)=k to get
(cid:0)
(cid:0)
p1
t
C
D
1
C
21
(cid:0)
2k.
1/k
(cid:0)
1 2k
k
(cid:0)
=k t k:
!
2
(cid:0)
1
(cid:0)
1
X
1
k
D
Replace t by
(cid:0)
4t and subtract from 1 to get
p1
1
(cid:0)
4t
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(cid:0)
1
(cid:0)
1
X
1
k
D
Replace t by
(cid:0)
4t and subtract from 1 to get
p1
1
(cid:0)
4t
(cid:0)
D (cid:0)
21
(cid:0)
2k.
1/k
(cid:0)
1 2k
(cid:0)
k
!
2
(cid:0)
1
(cid:0)
1
X
1
k
D
1/k22kt k
=k.
(cid:0)
2k
2
k
!
2
(cid:0)
1
(cid:0)
1
X
1
k
D
D
=k t k
(2.18)
(2.19)
1.
1/k
as .
(cid:0)
(cid:0)
and replace k
(cid:0)
1/k
.
1/2k
1
(cid:0)
.
1
1/(cid:0)
D
D
1 by k so that it varies from 0–
D (cid:0)
(cid:0)
(cid:0)
1 and 21
2k22k
(cid:0)
to get
2. Write .2k
2/ as 2.k
1/
(cid:0)
(cid:0)
D
(cid:0)
1
2k
2
k
!
=.k
C
1/ t k
1:
C
1
X
0
k
D
(2.20)
0(cid:140)1=.k
P1k
D
C
2n
n (cid:1) is called Catalan number. This can also be written as (cid:0)
k (cid:1)t k, which is the required OGF. The quan-
Now divide by 2t to get F .t/
1/(cid:141) (cid:0)
tity (cid:140)1=.n
1(cid:1)
C
2n
is always less than (cid:0)
n (cid:1) (as can be seen from Pascal’s triangle) this difference is always positive.
Moreover, both of them are integers showing that Catalan numbers are always integers. The
first few of them are 1;
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cid:1) (as can be seen from Pascal’s triangle) this difference is always positive.
Moreover, both of them are integers showing that Catalan numbers are always integers. The
first few of them are 1; 1; 2; 5; 14; 42; 132; 429; : : :.
1(cid:1). As (cid:0)
2n
(cid:0)
n
1/(cid:141) (cid:0)
2n
n (cid:1)
2n
n
D
(cid:0)
C
C
2k
r
Example 2.10 If m; n, and r are integers, prove the identity P
k
0 (cid:0)
m
k (cid:1)(cid:0)
r
k(cid:1)
n
(cid:0)
D
D
m
n
r (cid:1).
C
(cid:0)
Solution 2.10 As the LHS is the convolution of binomial coefficients, it suggests that it has
t/m
OGF of the form .1
n,
is the OGF of (cid:0)
which generates the RHS.
t/k, for some integer k. We have shown in the last chapter that .1
m
k (cid:1). Hence, the OGF of LHS is .1
t/n or equivalently .1
t/m.1
C
t/m
C
C
C
C
C
28
2. OPERATIONS ON GENERATING FUNCTIONS
2.1.8 DIFFERENTIATION AND INTEGRATION
A univariate GF can be differentiated or integrated in the dummy variable (which is assumed
to be continuous). Similarly, bivariate and higher GF can be differentiated or integrated in one
or more dummy variables, as they are independent. See Table 2.3 for a summary of various
operations.
Table 2.3: Summary of generating functions operations
Differentiation of OGF
Consider the OGF
F .t /
a0
C
D
a1t
C
a2t 2
C (cid:1) (cid:1) (cid:1) D
Differentiate w.r.t.
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F .t /
a0
C
D
a1t
C
a2t 2
C (cid:1) (cid:1) (cid:1) D
Differentiate w.r.t. t to get
akt k:
1
X
0
k
D
(2.21)
.@=@t /F .t/
a1
C
D
2a2t
C
3a3t 2
C (cid:1) (cid:1) (cid:1) D
.kak/t k
(cid:0)
1;
(2.22)
which is the OGF of (a1; 2a2; 3a3; : : :) or
follows:
If .a0; a1; a2; : : :/
F .t /, then .a1; 2a2; 3a3; : : :/
,
bn
f
D
.n
C
1/an
1
C
g
0. This can be stated as
(cid:21)
.@=@t /F .t /
D
,
F 0.t /. Literally, this means
1
X
1
k
D
for n
NameExpressionGFAdd/subtract∞ ak ± bktkF(t) ± G(t)Scalar multiply∞ cakc*F(t)Convolution∞ cktkF (t)*G(t)Left shift (EGF)∞ ak+1 tk/k!F ′(t) Left shift (OGF)∞ ak+1 /(k + 1) tk(F(t) − F(0))/tLeft m-shift (OGF)∞ am+k tk (F (t) − a0 − a1t … − am−1tm−1)/tmIndex multiply∞ kak−1 tkt F (t)Tail sum∞ ∞ aj tkF (t)/(1−t) Derivative∞ ak+1 tkF ′(t)k=0k=0k=0k=0k=0k=0k=0k=0j=0k=1 that differentiation of an OGF wrt the dummy variable multiplies each term by its index, and
shifts the whole sequence left one place.
Next consider an EGF
2.1. BASIC OPERATIONS
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that differentiation of an OGF wrt the dummy variable multiplies each term by its index, and
shifts the whole sequence left one place.
Next consider an EGF
2.1. BASIC OPERATIONS 29
H.t/
a0
C
D
a1t=1(cid:138)
C
a2t 2=2(cid:138)
C (cid:1) (cid:1) (cid:1) D
akt k=k(cid:138):
1
X
0
k
D
(2.23)
Differentiate w.r.t. t to get
.@=@t/H.t/
a1
C
D
a2t=1(cid:138)
C
a3t 2=2(cid:138)
C (cid:1) (cid:1) (cid:1) D
akt k
(cid:0)
1=.k
1/(cid:138);
(cid:0)
(2.24)
1
X
1
k
D
because k in the numerator cancels out with k(cid:138) in the denominator leaving a .k
that differentiation of an EGF is equivalent to “shift-left” by one position.
(cid:0)
1/(cid:138). This shows
Example 2.11 If F .t/ and H.t/ denote the OGF and EGF of a sequence (a0; a1; a2; : : :) find
the sequence whose OGF are, (i) tF 0.t/ and (ii) tH 0.t/.
Solution 2.11 The OGF of F 0.t/ is found above as (a1; 2a2; 3a3;
tF 0.t/
(cid:1) (cid:1) (cid:1)
. This can be written as a1t
2a2t 2
3a3t 3
a1t
). Multiplying this by t gives
a2.p2t/2
a3.31=3t/3
D
C
C
, in which the kth term is ak.k1=kt/k. Next consider the EGF. As shown above, H 0.t
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t/2
a3.31=3t/3
D
C
C
, in which the kth term is ak.k1=kt/k. Next consider the EGF. As shown above, H 0.t/ is
(cid:1) (cid:1) (cid:1)
the EGF of left-shifted sequence (a1; a2; a3; : : :). Multiply both sides by t to get
C (cid:1) (cid:1) (cid:1)
C
C
C
t.@=@t/H.t/
a1t
D
C
a2t 2=1(cid:138)
C
a3t 3=2(cid:138)
C (cid:1) (cid:1) (cid:1) D
ak=.k
(cid:0)
1/(cid:138)t k;
1
X
1
k
D
(2.25)
where we have associated 1=.k
(cid:0)
1/(cid:138) with the series term ak. This is the OGF of
ak=.k
f
1/(cid:138)
.
g
(cid:0)
Higher-Order Derivatives
Differentiation and integration can be applied any number of times. For example, consider the
0 t k. Take log of both sides and differentiate to get
formal power series f .t/
1=.1
P1k
D
t/2. Successive differentia-
the first derivative as f 0.t/=f .t/
t/ so that f 0.t/
t/n
tion gives f .n/.t/
(cid:0)
D
1. A direct consequence of this result is the GF
C
t/
D
1=.1
n(cid:138)=.1
1=.1
D
D
(cid:0)
(cid:0)
D
(cid:0)
n
C
k
k
!
1
t k
(cid:0)
1=.1
t/n:
(cid:0)
D
1
X
0
k
D
(2.26)
Higher-order derivatives are used to find factorial moments of discrete random variables in the
next chapter.
Example 2.
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t/n:
(cid:0)
D
1
X
0
k
D
(2.26)
Higher-order derivatives are used to find factorial moments of discrete random variables in the
next chapter.
Example 2.12 Use differentiation and shift operations to prove that .1
of the squares an
n2.
D
t/=.1
(cid:0)
C
t/3 is the GF
30
2. OPERATIONS ON GENERATING FUNCTIONS
t/2/
Solution 2.12 Consider the OGF F .t/
t/2
0.k
t=.1
D
(cid:0)
t/3. The RHS deriva-
.1
2=.1
1=t
C
D
1/2t k. Multiply LHS and RHS by t so that the t in the denominator of LHS
tive is P1k
D
cancels out and the power on the RHS becomes t k
P1k
D
1. Take log of LHS and differentiate to get .1=F .t//@F=@t
.1
1. Now RHS is the OGF of n2.
P1k
D
D
t/
D
(cid:0)
0.k
C
t/(cid:141), from which @F=@t
1/t k
C
t/=(cid:140)t.1
1/t k. Multiply by t to get
t/=(cid:140)t.1
C
C
.1=.1
0.k
D
C
D
C
(cid:0)
(cid:0)
(cid:0)
C
Example 2.13 (kth derivative of OGF) Find the GF of the sequence an
n
(cid:0)
m
m (cid:1) for m fixed.
C
D
m
C
Solution 2.13 Consider F .x/
xn
m
C
.1=m(cid:138)/.@=@x/m.1=.1
1=m(cid:138). Repeat
(cid:0)
m
1/.n
x//
(cid:0)
(cid:0)
2/ : : : nxn=m(cid:138). This
1=.1
(cid:0)
D
x/m
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Repeat
(cid:0)
m
1/.n
x//
(cid:0)
(cid:0)
2/ : : : nxn=m(cid:138). This
1=.1
(cid:0)
D
x/m
1.
C
(cid:0)
xn
m=m(cid:138). The first derivative
C
the differentiation m times
D
to get F .k/.x/
m
n
D
m (cid:1)xn. Thus,
C
(cid:0)
m/
is F 0.x/
.n
C
.n
m/.n
C
the GF is
D
C
can be written as
Example 2.14 (kth derivative of EGF) Prove that DkH.t/
D
0 an
P1n
D
C
kt n=n(cid:138).
t n
1/(cid:138)
a2
Solution 2.14 Consider
Take the derivative w.r.t.
n
Use derivative of
D
a0
a1t=1(cid:138)
a3t 3=3(cid:138)
H.t/
t of both sides. As a0 is a constant,
1 for each term to get H 0.t/
t n
(cid:0)
a2t 2=2(cid:138)
C
C
C
a1
akt k=k(cid:138)
.
C (cid:1) (cid:1) (cid:1) C
C (cid:1) (cid:1) (cid:1)
its derivative is zero.
a2t=1(cid:138)
a3t 2=2(cid:138)
D
C
(cid:0)
D
a3t=1(cid:138)
a4t 2=2(cid:138)
D
C (cid:1) (cid:1) (cid:1)
C
(cid:3)
C
1=.k
akt k
. Differentiate again (this time a1 being a constant vanishes)
(cid:0)
(cid:1) (cid:1) (cid:1) C
to get H 00.t/
this pro-
cess k times. All terms whose coe
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this time a1 being a constant vanishes)
(cid:0)
(cid:1) (cid:1) (cid:1) C
to get H 00.t/
this pro-
cess k times. All terms whose coefficients are below ak will vanish. What remains is
H .k/.t/
1t=1(cid:138)
. This can be expressed using the summation
notation introduced in Chapter 1 as H .k/.t/
k/(cid:138). Using the change
of indexvar introduced in Shanmugam and Chattamvelli [2016], this can be written as
H .k/.t/
0, all higher-order terms vanish except the
constant ak.
kt n=n(cid:138). Now if we put t
. Repeat
k ant n
P1n
D
P1n
D
2t 2=2(cid:138)
C (cid:1) (cid:1) (cid:1) C
akt k
k=.n
2=.k
C (cid:1) (cid:1) (cid:1)
C (cid:1) (cid:1) (cid:1)
0 an
2/(cid:138)
ak
ak
ak
D
D
C
D
C
D
C
C
(cid:0)
(cid:0)
C
C
C
(cid:0)
(cid:0)
2.1.9
INTEGRATION
Integrating F .t/
Z
D
x
a0
C
a1t
C
a2t 2
C (cid:1) (cid:1) (cid:1) D
P1k
D
0 akt k gives
F .t/dt
a0x
C
D
0
a1x2=2
a2x3=3
C
C (cid:1) (cid:1) (cid:1) D
.ak
1=k/ xk:
(cid:0)
X
1
k
(cid:21)
(2.27)
Divide both sides by x to get
x
.1=x/ Z
0
F .t/dt
.a0=1/
C
D
.a1=2/x
C
.a
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)
(2.27)
Divide both sides by x to get
x
.1=x/ Z
0
F .t/dt
.a0=1/
C
D
.a1=2/x
C
.a2=3/x2
C (cid:1) (cid:1) (cid:1)
(2.28)
which is the OGF of the sequence
ak=.k
f
1/
k
g
0.
(cid:21)
C
2.2. INVERTIBLE SEQUENCES 31
Next consider the EGF
H.t/
a0
C
D
a1t=1(cid:138)
C
a2t 2=2(cid:138)
a3t 3=3(cid:138)
C
C (cid:1) (cid:1) (cid:1) D
Integrate term by term to get
akt k=k(cid:138):
(2.29)
1
X
0
k
D
x
Z
0
H.t/dt
a0x
C
D
a1x2=.1(cid:138)
2/
(cid:3)
C
a2x3=.2(cid:138)
3/
(cid:3)
C (cid:1) (cid:1) (cid:1) D
Integration of EGF is equivalent to shift-right by one position.
akxk
C
1=.k
C
1/(cid:138):
(2.30)
1
X
0
k
D
n
Example 2.15 Evaluate the sum P
k
D
n
Solution 2.15 Consider the OGF P
k
be written as an integral .1=x/ R
x/n
we get (cid:140).1
1(cid:141)=(cid:140).n
1
C
C
(cid:0)
C
0 (cid:0)
n
k(cid:1)=.k
1/.
C
n
k(cid:1)=.k
0 (cid:0)
D
t/ndt. As the integral of .1
1/xk. With the help of above result, this can
t/n is .1
1/
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k(cid:1)=.k
0 (cid:0)
D
t/ndt. As the integral of .1
1/xk. With the help of above result, this can
t/n is .1
1/,
1=.n
t/n
C
C
C
C
C
x
0 .1
1/x(cid:141) as the result.
C
Example 2.16 Find the OGF of the sequence 1, 1/3, 1/5, 1/7, : : :.
x2
C
x3=3
1
D
1. Integrate the RHS term by term to get x
Solution 2.16 Consider the sequence F .x/
x2/(cid:0)
sired coefficients. As the LHS is R .1
terms A=.1
(cid:0)
C
1=2. Thus .1
.1=2/(cid:140)1=.1
(cid:0)
D
RHS to get 1
log.1
2 (cid:140)
x/
(cid:0)
log.a=b/. This is the OGF of the original sequence.
x2/(cid:0)
x/. This gives A
B
D
C
1=.1
C
1
2 log..1
B=.1
1
x/
x/(cid:141)
C
x2/(cid:0)
(cid:0)
C
log.1
x/
C
C
C
D
(cid:0)
(cid:0)
(cid:0)
, which has the de-
1dx, we use partial fractions to break this into two
C (cid:1) (cid:1) (cid:1)
x7=7
C (cid:1) (cid:1) (cid:1)
C
C
C
, which has OGF .1
x6
x4
C
x5=5
B
0, from which A
1 and A
D
x/(cid:141). Now integrate term-by-term on the
x//, using log.a/
log.b/
x/=.1
D
D
(cid:0)
B
(cid:0)
D
C
(cid:0)
INVERTIBLE SEQUENCES
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term-by-term on the
x//, using log.a/
log.b/
x/=.1
D
D
(cid:0)
B
(cid:0)
D
C
(cid:0)
INVERTIBLE SEQUENCES
2.2
Two GFs F .t/ and G.t/ are invertible if F .t/G.t/
1. The G.t/ is called the “multiplicative
inverse” of F .t/. Consider the sequence (a0; a1; : : :). Let G.x/ be the multiplicative-inverse with
1. This gives c0
coefficients (b0; b1; : : :), so that F .t/G.t/
1=a0. In
n
general, bn
0, it is invertible and coefficients can be calculated
k/=a0. If a0
k
recursively. This shows that a necessary condition for an OGF to be “invertible” is that the
constant term is nonzero.
a0b0, from which b0
1 akbn
(cid:0)
D
⁄
D (cid:0)
.P
D
D
D
D
Example 2.17 Find the inverse of the sequence .1; 1; 1; : : :/.
1. Similarly, a0b1
Solution 2.17 Let G.t/ be the inverse with coefficients (b0; b1; : : :). Then a0b0
b0
efficients are zeros. For example, in b2
Similarly, all higher coefficients are zeros. Hence, the inverse is .1;
b1
1
(cid:3)
(cid:3)
C
0 put b0
D
D
0 implies that 1
b1
D
1 and b1
D (cid:0)
1; 0; 0; : : :/.
0 or b1
a1b0
D (cid:0)
b0
D
C
D
C
C
1
1 implies that
D
1. All other co-
0.
1, so that b2
D
(cid:0)
32
2. OPERATIONS ON GENERATING FUNCTIONS
2.3 COMPOSITION OF GENERATING FUNCTIONS
If F .x/ and G.x/ are two
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-
0.
1, so that b2
D
(cid:0)
32
2. OPERATIONS ON GENERATING FUNCTIONS
2.3 COMPOSITION OF GENERATING FUNCTIONS
If F .x/ and G.x/ are two GFs, the composition is defined as F .G.x//. Note that F .G.x// is
not the same as G.F .x//. Consider F .x/
x/. Then F .G.x// is
1=.1
1=.1
x.1
(cid:0)
Consider a discrete branching process where the size of the nth generation is given by
0 Yk, where Yk is the offspring distribution of kth individual. If all Yk’s are assumed
P
x//. The general result of composition requires higher powers of GF.
x/ and G.x/
x.1
Xn
k
D
C
C
D
(cid:0)
1
Xn
to be IID, it is easy to see that
(cid:0)
D
D
P (cid:140)Xn
k
j
D
Xn
(cid:0)
1
D
j (cid:141)
D
P (cid:2)Y1
Y2
C
C (cid:1) (cid:1) (cid:1) C
Yj (cid:3) :
Using total probability law of conditional probability, this becomes
P (cid:140)Xn
k(cid:141)
D
D
1
X
0
j
D
P (cid:2)Y1
Y2
C
C (cid:1) (cid:1) (cid:1) C
Yj (cid:3) P (cid:140)Xn
(cid:0)
1
D
j (cid:141) :
Let (cid:30)n.t/ denote the PGF of the size of nth generation. Then
(cid:30)n.t/
D
P .Xn
D
k/ t k:
1
(cid:0)
Xn
X
0
k
D
1.(cid:30).t//.
From this it follows that (cid:30)n.t/
(cid:30)n
(cid:0)
D
2.4
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Xn
X
0
k
D
1.(cid:30).t//.
From this it follows that (cid:30)n.t/
(cid:30)n
(cid:0)
D
2.4
SUMMARY
(2.31)
(2.32)
(2.33)
This chapter introduced various operations on GFs. This includes arithmetic operations, linear
combinations, left and right shifts, convolutions, powers, change of dummy variables, differen-
tiation and integration. These are applied in various problems in subsequent chapters.
C H A P T E R 3
33
Generating Functions in
Statistics
After finishing the chapter, readers will be able to
(cid:1) (cid:1) (cid:1)
• explain the probability generating functions.
• describe the CDF generating functions.
• explore moment and cumulant generating functions.
• comprehend characteristic functions.
• acquaint with mean deviation generating functions.
• discuss factorial moment generating functions.
3.1 GENERATING FUNCTIONS IN STATISTICS
GFs are used in various branches of statistics like distribution theory, stochastic processes, etc. A
one-to-one correspondence is established between the power series expansion of a GF in one or
more auxiliary (dummy) variables, and the coefficients of a known sequence. These coefficients
differ in various GFs as shown below. GFs used in discrete distribution theory are defined on the
sample space of a random variable or probability distribution. They can be finite or infinite, de-
pending on the corresponding distribution. Multiple GFs like PGF, MGF, etc. can be defined
on the same random variable, as these generate different quantities as shown in the follow-
ing paragraphs. Some of these GFs can be obtained from others using various transformations
discussed in Chapter 2. The GF technique is especially useful in some distributions in inves-
tigating inter-relationships and asymptotics. As an example, consider the Poisson distribution
with parameter (cid:21). If the average number of occurrences of a rare event in a certain time interval
dt/ is (cid:21), the Poisson probabilities gives us the chance of observing exactly k events in the
.t; t
same time interval, if various events occur independently of each other. The PGF of Poisson
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dt/ is (cid:21), the Poisson probabilities gives us the chance of observing exactly k events in the
.t; t
same time interval, if various events occur independently of each other. The PGF of Poisson
distribution is infinitely differentiable, and it is easy to find factorial moments because the kth
derivative of PGF is (cid:21)k times the PGF. The GFs can be used to prove the additivity property
of Poisson, negative binomial, and many other distributions [Shanmugam and Chattamvelli,
C
34
3. GENERATING FUNCTIONS IN STATISTICS
2016]. Although the previous chapter used x as the dummy variable in a GF, we will use t as
the dummy variable in this chapter because x is regarded as a random variable, and used as a
subscript.
3.1.1 TYPES OF GENERATING FUNCTIONS
There are four popular GFs used in statistics—namely (i) probability generating function (PGF),
denoted by Px.t/; (ii) moment generating function (MGF), denoted by Mx.t/; (iii) cumulant
generating function (CGF), denoted by Kx.t/; and (iv) characteristic function (ChF), denoted
by (cid:30)x.t/. In addition, there are still others like factorial moment GF (FMGF), inverse moment
GF (IMGF), inverse factorial moment GF (IFMGF), absolute moment GF, as well as GFs
for odd moments and even moments separately. These are called “canonical functions” in some
fields.
The PGF generates the probabilities of a random variable, and is of type OGF. The rising
and falling FMGFs are also of type OGF. The MGF generates moments, and is of type EGF.
It has further subdivisions as ordinary MGF, central MGF, FMGF, IMGF, and IFMGF. The
CGF and ChF are also related to MGF. Some of these (MGF, ChF, etc.) can also be defined
ln.Mx.t//, which
for an arbitrary origin. The CGF is defined in terms of the MGF as Kx.t/
when expanded as a polynomial in t gives the cumulants. Note that the logarithm
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ln.Mx.t//, which
for an arbitrary origin. The CGF is defined in terms of the MGF as Kx.t/
when expanded as a polynomial in t gives the cumulants. Note that the logarithm is to the base
e.ln/. As every distribution does not possess a MGF, the concept is extended to the complex
domain by defining the ChF as (cid:30)x.t/
E.eitx/. If all of them exists for a distribution, then
D
D
Px.et /
Mx.t/
eKx .t/
(cid:30)x.it/:
(3.1)
D
This can also be written in the alternate forms Px.eit /
Px.t/
D
(cid:30)x.i ln.t// (see Table 3.1).
Mx.ln.t//
eKx .ln.t//
D
D
D
D
D
Mx.it/
eKx .it/
D
(cid:30)x.
(cid:0)
D
t/ or as
3.2
PROBABILITY GENERATING FUNCTIONS (PGF)
The PGF is extensively used in statistics, econometrics, and various engineering fields. It is a
compact mathematical expression in one or more dummy variables, along with the unknown
parameters, if any, of a distribution.
Definition 3.1 A GF in which the coefficients are the probabilities associated with a random
variable is called the PGF.
As mentioned in Chapter 1, the PGF is not a function that generates probabilities when
particular values are plugged in for the dummy variable t. But when expanded as a power series
in the auxiliary variable, the coefficient of t k is the probability of the random variable X taking
the value k. This is one way the PGF of a random variable can be used to generate probabilities.
Table 3.1: Summary table of generating functions
3.2. PROBABILITY GENERATING FUNCTIONS (PGF)
35
It is defined as
Px.t/
D
E.t x/
X
x
D
t xp.x/
p.0/
C
D
p.1/ t
C
p.2/ t 2
C (cid:1) (cid:1) (cid:1) C
p.k/ t k
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.x/
p.0/
C
D
p.1/ t
C
p.2/ t 2
C (cid:1) (cid:1) (cid:1) C
p.k/ t k
C (cid:1) (cid:1) (cid:1)
;
(3.2)
where the summation is over the range of X. This means that the PGF of a discrete random
variable is the weighted sum of all probabilities of outcomes k with weight t k for each value
of k in its range. Obviously the RHS is a finite series for distributions with finite range. It
usually results in a compact closed form expression for discrete distributions. It converges for
< 1, and appropriate derivatives exist. Differentiating both sides of (3.2) k times w.r.t. t gives
t
j
.@k=@t k/Px.t/
0, all higher order terms that have “t”
terms involving t. If we put t
or higher powers vanish, giving k(cid:138)p.k/. From this p.k/ is obtained as .@k=@t k/Px.t/
0=k(cid:138). If
the Px.t/ involves powers or exponents, we take the log (w.r.t. e) of both sides and differentiate
1/ to simplify the differentiation. The PGF
k times, and then use the following result on Px.t
k(cid:138)p.k/
C
D
D
D
j
j
t
D
AbbreviationSymbolDefi nitionE=exp oper.GeneratesWhatHow Obtained(t = dummy-variable)PGFPx(t)E(tx)Probabilitiespk = ∂k Px(t)|t=0/k!CDFGFFx(t)E(tx/(1– t))Cumulative probabilitiesFk = 1 ∂k Px(t)/(1– t)|t=0MGFMx(t)E(etx)Momentsµk = ∂k Mx(t)|t=0CMGFMz(t)E(et(x-μ))Central momentsµk = ∂k Mz(t)|t=0ChFφx(t)E(eitx)Momentsik µk = ∂k φx(t)|t=0CG
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(et(x-μ))Central momentsµk = ∂k Mz(t)|t=0ChFφx(t)E(eitx)Momentsik µk = ∂k φx(t)|t=0CGFKx(t)log(E(etx))Cumulantsµk = ∂k Kx(t)|t=0FMGFΓx(t)E(1 + t)x)Factorial momentsµ(k) = ∂k Γx(t)|t=0MDGFDx(t)2E(tx/(1 – t)2)Mean deviationSee Section 3.5Probability generating function (PGF) is of type OGF. MGF and ChF are of type EGF. MGF need not al-ways exist, but characteristic function always exists. Falling factorial moment is denoted as µ(k) = E(x(x – 1)(x − 2) … (x − k + 1)).∂tk∂tk∂tk∂tk∂tk∂tkk! ∂tk 36
3. GENERATING FUNCTIONS IN STATISTICS
is immensely useful in deriving key properties of a random variable easily. As shown below, it is
related to CDFGF and MDGF.
Example 3.1 (PGF special values Px.t
from the PGF of a discrete distribution.
D
0/ and Px.t
D
1/) Find Px.t
0/ and Px.t
1/
D
D
Solution 3.1 As Pk p.k/ being the sum of the probabilities is one, it follows trivially by putting
p.0/, the first probabil-
t
(cid:141).
ity. Similarly, put t
1. Put t
D
1 to get the RHS as (cid:140)p.0/
0 in (3.2) to get Px.t
1 in (3.2) that Px.t
D
(cid:140)p.1/
p.2/
p(cid:140)3(cid:141)
p(cid:140)5(cid:141)
D
(cid:141)
0/
1/
D
D
D
C
C (cid:1) (cid:1) (cid:1) C
(cid:0)
C
C
C (cid:1) (cid:1) (cid:
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1/
D
D
D
C
C (cid:1) (cid:1) (cid:1) C
(cid:0)
C
C
C (cid:1) (cid:1) (cid:1)
D (cid:0)
Example 3.2 (PGF of arbitrary distribution) Find the PGF of a random variable given below,
and obtain its mean.
x
p.x/
2
1/3
3
1/6
1
1/2
Solution 3.2 Plug in the values directly in
Px.t/
D
E.t x/
D
t xp.x/
3
X
1
x
D
to get Px.t/
2t=3
D
3t 2=6
t
j
C
t=2
C
D
1
D
t 2=3
1=2
C
C
2=3
3=6
C
D
5=3.
t 3=6. Differentiate w.r.t. t and put t
(3.3)
1 to get the mean as 1=2
C
D
Example 3.3 (PGF of uniform distribution) Find the PGF of a uniform distribution, and ob-
tain the difference between the sum of even and odd probabilities. Also obtain the mean.
Solution 3.3 Take f .x/
1=k for x
1; 2; : : : k for simplicity. Then Px.t/
D
t k(cid:141). Take t outside, and apply formula for geometric progression to get Px.t/
D
D
1=k(cid:140)t 1
t 2
C
.t=k/.t k
(cid:1) (cid:1) (cid:1) C
1/=.t
1/. Take log and differentiate. Then put t
0, to get the mean as .k
C
D
(cid:0)
C
(cid:0)
D
1/=2.
Example 3.4 (PGF of Poisson distribution) Find the PGF of a Poisson distribution, and ob-
tain the difference between the sum of even and odd probabilities. Also obtain the kth moment.
Solution 3.4 The PGF of a Poisson distribution is
Px.t/
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F of a Poisson distribution, and ob-
tain the difference between the sum of even and odd probabilities. Also obtain the kth moment.
Solution 3.4 The PGF of a Poisson distribution is
Px.t/
D
E.t x/
D
1
X
0
x
D
t xe(cid:0)
(cid:21)(cid:21)x=x(cid:138)
e(cid:0)
D
(cid:21) 1
X
0
x
D
.(cid:21)t/x=x(cid:138)
e(cid:0)
(cid:21)et(cid:21)
D
D
e(cid:0)
(cid:21)(cid:140)1
(cid:0)
t(cid:141):
(3.4)
Put t
exp.
D (cid:0)
2(cid:21)/.
(cid:0)
1 in (3.4) and use the above result to get the desired sum as exp.
(cid:21)(cid:140)1
.
(cid:0)
(cid:0)
1/(cid:141)/
(cid:0)
D
3.2. PROBABILITY GENERATING FUNCTIONS (PGF)
37
Example 3.5 (PGF of geometric distribution) Find the PGF of a geometric distribution with
0; 1; 2 : : :, and obtain the difference between the sum of even and
PMF f .x
D
odd probabilities.
qxp; x
p/
D
I
Solution 3.5 As x
0; 1; 2; : : :
1
D
values, we get the PGF as
Px.t/
D
E.t x/
D
1
X
0
x
D
t xqxp
p
D
1
X
0
x
D
q4
.qt/x
p=.1
qt /:
(cid:0)
D
(3.5)
q2
q2/
q0p
q2p
C
C
C
D
p(cid:140)1
p=.1
C
q2
C
q4
D
q1p
C
q3p
C (cid:1) (cid:1) (
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D
p(cid:140)1
p=.1
C
q2
C
q4
D
q1p
C
q3p
C (cid:1) (cid:1) (cid:1) D
C (cid:1) (cid:1) (cid:1)
(cid:141)
D
C (cid:1) (cid:1) (cid:1) D
qp(cid:140)1
(cid:141)
D
qp=.1
Now P [X is even]
(cid:0)
q2/
P [X is odd]
the above result, the difference between these must equal the value of Px.t
in (3.5) to get p=.1
q/, which is the same as 1=.1
p=.1
p=.1
f .x
D
D (cid:0)
q/
C
q/. There is another version of geometric distribution with range 1–
C
D
with PMF
C
1p [Shanmugam and Chattamvelli, 2016]. In this case the PGF is pt=.1
p/
qt/.
Closed-form expressions for Px.t/ are available for most of the common discrete dis-
tributions. They are seldom used for continuous distributions because R t xf .x/dx may not be
convergent.
q/, and
C
q/. Using
1
C
1/. Put t
q=.1
D
q=.1
D (cid:0)
q/
(cid:0)
1
C (cid:1) (cid:1) (cid:1)
1=.1
1//
qx
q.
D
C
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
I
Example 3.6 (PGF of BINO.n; p/ distribution) Find the PGF of BINO.n; p/ distribution
x, and obtain the mean.
with PMF f .x
n; p/
n
x(cid:1)pxqn
(cid:0)
(cid:0)
I
D
Solution 3.6 By definition
Px.t/
D
E.t x/
D
!
pxqn
(cid:0)
xt x
n
x
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(cid:0)
(cid:0)
I
D
Solution 3.6 By definition
Px.t/
D
E.t x/
D
!
pxqn
(cid:0)
xt x
n
x
n
X
0
x
D
!
.pt/xqn
n
x
n
X
0
x
D
D
x
(cid:0)
.q
C
D
pt/n:
(3.6)
The coefficient of t x gives the probability that the random variable takes the value x. To find the
mean, we take log of both sides. Then log.Px.t//
pt/. Differentiate both sides
w.r.t. t to get P 0x.t/=Px.t/
1 to get the
RHS as n
n
(cid:3)
pt/. Now put t
1.
n
D
(cid:3)
np as q
1 and use Px.t
p=.q
p
log.q
p=.q
p/
1/
D
D
C
D
D
(cid:3)
C
D
C
C
D
Example 3.7 (PGF of NBINO.n; p/ distribution) Find the PGF of negative binomial distri-
bution NBINO.n; p/, and obtain the mean.
Solution 3.7 By definition
Px.t/
D
E.t x/
D
1
X
0
x
D
x
C
k
x
(cid:0)
!
1
pkqxt x
x
!
1
C
k
x
(cid:0)
D
1
X
0
x
D
pk.qt/x:
(3.7)
38
3. GENERATING FUNCTIONS IN STATISTICS
As pk is a constant, this is easily seen to be (cid:140)p=.1
r/ and q
and put t
get another form of negative binomial distribution NBINO.n; 1=.1
r/(cid:141)k(cid:140)r=.1
C
by putting x
r/(cid:141)x with PGF 1=(cid:140)1
t .1
(cid:0)
p/ in the expansion 1=.1
kq=p. By setting p
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putting x
r/(cid:141)x with PGF 1=(cid:140)1
t .1
(cid:0)
p/ in the expansion 1=.1
kq=p. By setting p
1 to get E.X /
(cid:0)
D
r.1
tq
D
C
D
C
qt/(cid:141)k. Take log and differentiate w.r.t. t
r/ we
1
1=.1
p
D
(cid:0)
x
r// as (cid:0)
C
(cid:1)(cid:140)1=.1
r=.1
1
k
x
C
(cid:0)
D
C
C
t /(cid:141)k. The PGF of NBINO.n; p/ can directly be derived
x/n and multiplying both sides by pn.
D
D
(cid:0)
(cid:0)
Example 3.8 (PGF of logarithmic distribution) Find the PGF of logarithmic.q/ distribution,
and obtain the mean.
Solution 3.8 By definition
Px.t/
D (cid:0)
1
X
1
x
D
qxt x=.x log.p//
1= log.p//
.
(cid:0)
D
1
X
1
x
D
.qt /x=x
log.1
(cid:0)
D
qt/= log.1
(cid:0)
q/;
(3.8)
1
D
where q
p and t < 1=q for convergence. This distribution is a special limiting case of
negative binomial distribution when the zero class is dropped, and the parameter k of negative
binomial distribution
0.
(cid:0)
!
hkpk=.1
(cid:0)
pk/i (cid:2)0; q; .k
1/q2=2(cid:138)
.k
C
C
1/.k
C
2/q3=3(cid:138)
(cid:3) :
C (cid:1) (cid:1) (cid:1)
C
(3.9)
D
!
0, a factorial term will remain in the numerator which cancels out with the denom-
When k
1/.k
inator, leaving a k in the denominator
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1) (cid:1)
C
(3.9)
D
!
0, a factorial term will remain in the numerator which cancels out with the denom-
When k
1/.k
inator, leaving a k in the denominator. For example, .k
C
q3=3. Divide the numerator and denominator of the term outside the
2/q3=3(cid:138)
1:2q3=3(cid:138)
bracket by pk to get k=p(cid:0)
1=.ln.p//.
Thus, the negative binomial distribution tends to the logarithmic law in the limit. Differentiate
the PGF w.r.t. t to get @[email protected] /
1 to get the mean as
q=(cid:140)p log.p/(cid:141).
q=(cid:140)log.p/.1
q/(cid:141)
1. Use L’Hospitals rule once to get this limit as
2/q3=3(cid:138) becomes .0
qt /. Put t
1= log.p//
q=.1
1/.0
C
D
D
C
C
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
.
k
(cid:0)
D
Example 3.9 (PGF of hypergeometric distribution) Find the PGF of a hypergeometric distri-
N
n (cid:1), and obtain the difference between the sum of even and odd probabilities.
bution (cid:0)
m
k (cid:1)(cid:0)
N
n
m
k (cid:1)=(cid:0)
(cid:0)
(cid:0)
Solution 3.9 The PGF is easily obtained as
Px.t /
D
m
X
0
x
D
m
x
! N
n
(cid:0)
(cid:0)
!
=
m
x
!
t x
N
n
(3.10)
because the probabilities outside the range .x > m/ are all zeros. The hypergeometric series
a.k/b.k/
xk
k(cid:138) where a.k/ denotes rising Pochhammer
is defined in terms of as F .x
c.k/
number.
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zeros. The hypergeometric series
a.k/b.k/
xk
k(cid:138) where a.k/ denotes rising Pochhammer
is defined in terms of as F .x
c.k/
number. A property of the hypergeometric series is that the ratio of two consecutive terms is
a; b/
Pk
D
I
3.2. PROBABILITY GENERATING FUNCTIONS (PGF)
39
1/th to kth term is a polynomial in k. If such
a polynomial. More precisely, the ratio of .k
C
a polynomial is given, we could easily ascertain the parameters of the hypergeometric function
from it. Consider the ratio of two successive terms of hypergeometric distribution
f .k
C
1/=f .k/
.k
(cid:0)
D
m/.k
(cid:0)
n/=(cid:140).k
N
m
n
(cid:0)
(cid:0)
C
C
1/.k
C
1/(cid:141)
(3.11)
which is always written with “k” as the first variable. Now drop each k, and collect the remain-
ing values to find out the corresponding hypergeometric function parameters as F .
m
x/. From the PMF we have that f .0/
plier of the hypergeometric function to get f .x/
n;
From this we get the PGF as (cid:0)
(cid:0)
N
m
n (cid:1)=(cid:0)
n (cid:1). This should be used as multi-
(cid:0)
N
m
x/.
n (cid:1)F .
n (cid:1)=(cid:0)
(cid:0)
n
m
N
n
I
(cid:0)
t/.
n
I
1
I
1
I
m;
C
C
m
N
N
(cid:0)
(cid:0)
(cid:0)
(cid:0)
(cid:0)
n
n
N
N
m
n (cid:1)=(cid:0)
(cid:0)
N
n (cid:1)2F1.
(cid:0)
N
(cid:0)
m
I
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:
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0)
(cid:0)
N
n (cid:1)2F1.
(cid:0)
N
(cid:0)
m
I
D
D
(cid:0)
(cid:0)
(cid:0)
(cid:0)
m;
1
I
(cid:0)
C
Example 3.10 (PGF of power-series distribution) Find the PGF of power-series distribution,
and obtain the mean and variance.
Solution 3.10 The PMF of power-series distribution is P .X
any integer values. By definition
k/
D
D
ak(cid:18) k=F .(cid:18)/ where k takes
Px.t/
D
.1=F .(cid:18)//
1
X
1
x
D
ax(cid:18) xt x
D
.1=F .(cid:18)//
ax.(cid:18) t/x
D
1
X
1
x
D
F .(cid:18) t/=F .(cid:18) /:
(3.12)
Differentiate w.r.t. t to get P 0x.t/
(cid:18) F 0.(cid:18)/=F .(cid:18)/. The variance is found using V .X /
F 00.t/
D
(cid:18) 2F 00.(cid:18)/=F .(cid:18)/
E.X/
(cid:18) 2F 00.(cid:18) t/=F .(cid:18)/. From this the variance is obtained as F 00.1/
(cid:140)(cid:18)F 0.(cid:18)/=F .(cid:18) /(cid:141)2.
(cid:18) F 0.(cid:18) t/=F .(cid:18)/, from which the mean (cid:22)
(cid:18)F 0.(cid:18)/=F .(cid:18)/
E.X.X
(cid:0)
C
1//
D
D
C
(cid:0)
t
1
D
P 0x.t/
D
(cid:140
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F .(cid:18)/
E.X.X
(cid:0)
C
1//
D
D
C
(cid:0)
t
1
D
P 0x.t/
D
(cid:140)E.X /(cid:141)2. Consider
F 0.1/
(cid:140)F 0.1/(cid:141)2
D
j
D
(cid:0)
C
(cid:0)
3.2.1 PROPERTIES OF PGF
1. P .r/.0/=r(cid:138)
@r =@t r Px.t/
P (cid:140)X
Px.t/ is infinitely differentiable in t for
@r
@t r Px.t/
D
D
D
0
j
t
D
r(cid:141).
< 1. Differentiate Px.t/
D
t
j
j
E.t x/r times to get
D
E (cid:140)x.x
(cid:0)
1/ : : : .x
r
(cid:0)
C
1/t x
r (cid:141)
(cid:0)
D
X
r
x
(cid:21)
(cid:140)x.x
(cid:0)
1/ : : : .x
r
(cid:0)
C
1/t x
(cid:0)
r (cid:141) f .x/:
(3.13)
The first term in this sum is (cid:140)r.r
r(cid:138)f .x
r(cid:138)f .x
r/. By putting t
r/. Thus, @r
D
@t r Px.t
0/
D
D
D
D
r(cid:138)f .x
r/.
D
1/t r
D
0, every term except the first vanishes, and the RHS becomes
(cid:140)r(cid:138)t 0(cid:141)f .x/
1/ : : : .r
r (cid:141)f .x
r/
D
D
C
(cid:0)
(cid:0)
r
(cid:0)
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(cid:141)f .x/
1/ : : : .r
r (cid:141)f .x
r/
D
D
C
(cid:0)
(cid:0)
r
(cid:0)
2. P .r/.1/=r(cid:138)
E(cid:140)X.r/(cid:141), the r th falling factorial moment (page 56). By putting t
r
1
1/(cid:141), which is the r th falling facto-
in (3.13), the RHS becomes E(cid:140)x.x
rial moment. This is sometimes called the FMGF (see Section 3.10 on page 56).
1/ : : : .x
D
C
D
(cid:0)
(cid:0)
40
3. (cid:22)
E.X/
3. GENERATING FUNCTIONS IN STATISTICS
P 0.1/, and (cid:22)02 D
P 0.1/.
The first result follows directly from the above by putting r
the second result also follows from it.
E.X 2/
P 00.1/
D
D
C
D
1. As X 2
D
X.X
1/
(cid:0)
C
X,
D
4. V(X) = P 00.1/
P 0.1/(cid:140)1
This result follows from the fact that V .X/
E(cid:140)X(cid:141)2. Now use the above two results.
P 0.1/(cid:141).
C
(cid:0)
E(cid:140)X 2(cid:141)
E(cid:140)X(cid:141)2
(cid:0)
E(cid:140)X.X
1/(cid:141)
(cid:0)
C
E(cid:140)X(cid:141)
(cid:0)
D
D
1
5. Rt Px.t/dt
D
D
t
j
Consider Rt Px.t/dt
integrate to get the result. This is the first inverse moment (of X
random variables.
E. 1
1 /.
X
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.t/dt
D
D
t
j
Consider Rt Px.t/dt
integrate to get the result. This is the first inverse moment (of X
random variables.
E. 1
1 /.
X
C
Rt E.t x/dt. Take expectation operation outside the integral and
D
1), and holds for positive
C
6. PcX .t/
PX .t c/.
D
This follows by writing t cX as .t c/X .
7. PX
c.t/
t (cid:6)
This follows by writing t X
Px.t/.
D
(cid:3)
(cid:6)
c
c as .t (cid:6)
c/t X .
(cid:6)
8. PcX
d .t/
(cid:6)
d
t (cid:6)
PX .t c/.
D
(cid:3)
This follows by combining two cases above.
9. Px.t/
MX .ln.t//.
From (3.1), we have Px.et /
D
10. P.X
(cid:22)/=(cid:27) .t/
(cid:6)
t (cid:6)
D
D
(cid:22)=(cid:27) PX .t 1=(cid:27) /.
Mx.t/. Write t 0 D
et so that t
D
ln.t 0/ to get the result.
This is called the change of origin and scale transformation of PGF. This follows by com-
bining (6) and (7).
11. If X and Y are IID random variables, P.X
Px.t/PY .
t/.
(cid:0)
Y /.t/
C
D
Px.t/Py.t/ and P.X
Y /.t/
(cid:0)
D
3.3 GENERATING FUNCTIONS FOR CDF
As the PGF of a random variable generates probabilities, it can be used to generate the sum of
left tail probabilities (CDF).
Definition 3.2 A GF in which the coefficients are the cumulative probabilities associated with
a random variable is called the CDF generating function (CDFGF).
We have seen in Theorem 1.1 (page 9) that if F .x/ is the OGF of the sequence
(a0; a1; : : : ; an; : : :), fi
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