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Generating Functions in Engineering and the Applied Sciences

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Generating function (GF) is a mathematical technique to concisely represent a known ordered sequence into a simple continuous algebraic function in dummy variable(s). This Second Edition introduces commonly encountered generating functions (GFs) in engineering and applied sciences, such as ordinary GF (OGF), exponential GF (EGF), as also Dirichlet GF (DGF), Lambert GF (LGF), Logarithmic GF (LogGF), Hurwitz GF (HGF), Mittag-Lefler GF (MLGF), etc.  This book is intended mainly for beginners in applied science and engineering fields to help them understand single-variable GFs and illustrate how to apply them in various practical problems.  Specifically, the book discusses probability GFs (PGF),  moment and cumulant GFs (MGF, CGF), mean deviation GFs (MDGF), survival function GFs (SFGF), rising and falling factorial GFs, factorial moment, and inverse factorial moment GFs.  Applications of GFs in algebra, analysis of algorithms, bioinformatics, combinatorics, economics, finance, genomics, geometry, graph theory, management, number theory, polymer chemistry, reliability, statistics and structural engineering have been added to this new edition. This book is written in such a way that readers who do not have prior knowledge of the topic can easily follow through the chapters and apply the lessons learned in their respective disciplines.

Author(s): Rajan Chattamvelli, Ramalingam Shanmugam
Series: Synthesis Lectures on Engineering, Science, and Technology
Edition: 2
Publisher: Springer
Year: 2023

Language: English
Pages: 128
City: Cham

Preface
Contents
About the Authors
Abbreviations
1 Types of Generating Functions
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1.1 Introduction
1.1.1 Origin of Generating Functions
1.1.2 Existence of Generating Functions
1.2 Notations and Nomenclatures
1.2.1 Rising and Falling Factorials
1.2.2 Dummy Variable
1.3 Ordinary Generating Functions
1.3.1 Recurrence Relations
1.3.2 Types of Sequences
1.3.3 OGF for Partial Sums
1.4 Exponential Generating Functions (EGF)
1.5 Pochhammer Generating Functions
1.5.1 Rising Pochhammer GF (RPGF)
1.5.2 Falling Pochhammer GF (FPGF)
1.6 Dirichlet Generating Functions
1.7 Other Generating Functions
1.7.1 Lambert Generating Function
1.7.2 Exponentio-Ordinary GF
1.7.3 Logarithmic GF
1.7.4 Special Functions as Generating Function
1.7.5 Auto-covariance Generating Function
1.7.6 Information Generating Function (IGF)
1.8 Generating Functions in Number Theory
1.8.1 Lah Numbers
1.8.2 Rook Polynomial Generating Function
1.8.3 Stirling Numbers
1.9 Summary
2 Operations on Generating Functions
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2.1 Basic Operations
2.1.1 Extracting Coefficients
2.1.2 Addition and Subtraction
2.1.3 Multiplication by Non-Zero Constant
2.1.4 Linear Combination
2.1.5 Shifting
2.1.6 Functions of Dummy Variable
2.1.7 Convolutions and Powers
2.1.8 Differentiation and Integration
2.2 Multiplicative Inverse Sequences
2.3 Composition of Generating Functions
2.4 Summary
3 Generating Functions in Statistics
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3.1 Uses of Generating Functions in Statistics
3.1.1 Types of Generating Functions
3.2 Probability Generating Functions (PGF)
3.2.1 Properties of PGF
3.3 Generating Functions for CDF
3.3.1 Properties of CDFGF
3.4 Generating Function for Survival Probabilities
3.5 Generating Functions for Mean Deviation
3.5.1 Properties of MDGF
3.6 MD of Some Distributions
3.6.1 MD of Geometric Distribution
3.6.2 MD of Binomial Distribution
3.6.3 MD of Poisson Distribution
3.6.4 MD of Negative Binomial Distribution
3.6.5 MD of Logarithmic Distribution
3.7 Moment Generating Functions (MGF)
3.7.1 Properties of Moment Generating Functions
3.8 Characteristic Functions
3.8.1 Properties of Characteristic Functions
3.9 Cumulant Generating Functions
3.9.1 Relations Among Moments and Cumulants
3.10 Factorial Moment Generating Functions
3.11 Conditional Moment Generating Functions (CMGF)
3.12 Generating Functions of Truncated Distributions
3.13 Convergence of Generating Functions
3.14 Summary
4 Applications of Generating Functions
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4.1 Applications in Algebra
4.1.1 Series Involving Integer Parts
4.1.2 Permutation Inversions
4.1.3 Generating Function of Strided Sequences
4.2 Applications in Geometry
4.3 Applications in Computing
4.3.1 Well-Formed Parentheses
4.3.2 Merge-Sort Algorithm Analysis
4.3.3 Quick-Sort Algorithm Analysis
4.3.4 Binary-Search Algorithm Analysis
4.3.5 Formal Languages
4.4 Applications in Combinatorics
4.4.1 Combinatorial Identities
4.4.2 New Generating Functions from Old
4.4.3 Recurrence Relations
4.4.4 Towers of Hanoi Puzzle
4.5 Applications in Structural Engineering
4.6 Applications in Graph Theory
4.6.1 Graph Enumeration
4.6.2 Tree Enumeration
4.7 Applications in Chemistry
4.7.1 Polymer Chemistry
4.7.2 Counting Isomers of Hydrocarbons
4.7.3 Modeling Polymerization
4.8 Applications in Epidemiology
4.9 Applications in Number Theory
4.10 Applications in Statistics
4.10.1 Sums of IID Random Variables
4.10.2 Infinite Divisibility
4.10.3 Applications in Stochastic Processes
4.11 Applications in Reliability
4.11.1 Series-Parallel Systems
4.12 Applications in Bioinformatics
4.12.1 Lifetime of Cellular Proteins
4.12.2 Sequence Alignment
4.13 Applications in Genetics
4.14 Applications in Management
4.14.1 Annuity
4.15 Applications in Economics
4.16 Miscellaneous Applications
4.17 Summary
Index
Index