Theory and Applications of the Generating Functions: Pure Mathematics and Applied Science

The generating functions have various applications in many branches of mathematics and sciences, representing a widely used and powerful tool for solving problems. In combinatorics, they allow for obtaining a compact representation of discrete structures and the investigation of several properties of the sequences they generate, i.e. their asymptotic growth. Theory and Applications of the Generating Functions: Pure Mathematics and Applied Science book provides the mathematical basis and application examples: generating functions, infinite series, and asymptotic methods. Cover Half Title Page Title Page Copyright Declaration About the Editor Table of Contents List of Contributors List of Abbreviations Preface Section 1: Generating Functions And Series Chapter 1: Generating Function for the Figurative Numbers of Regular Polyhedron Abstract Introduction Materials and Methods Results and Discussion Main Text Conclusion Data Availability Conflicts of Interest References Chapter 2: Method for Obtaining Coefficients of Powers of Bivariate Generating Functions Abstract Introduction Composita of A Multivariate Generating Function Operations on Compositae of Bivariate Generating Functions Application of Compositae for Obtaining Coefficients of Bivariate Generating Functions Conclusions References Chapter 3: Danny Nguyen and Igor Pak Abstract Introduction Notations Polynomial Time Operations on Short Gfs Short Gfs and the Class P/poly Short Gfs and the Non-uniform Polynomial Hierarchy A Hierarchy of Generating Function Short GFS Have Long Projection Intersections, Unions and Minkowski Sums of Short Gfs Squares, Primes, and Short GFS Relative Complexity of Short GF Proof of Lemma 4.10 Final Remarks and Open Problems Acknowledgements References Chapter 4: Generating Functions for Some Hypergeometric Functions of Four Variables Via Laplace Integral Representations Abstract Introduction Generating Relations Special Cases Conclusion Acknowledgments References Chapter 5: Identities, Inequalities for Boole-type Polynomials: Approach to Generating Functions and Infinite Series Abstract Introduction Generating Functions for Peters Type Combinatorial Numbers And Polynomials Inequalities for Stirling Numbers of the Second Kind and Finite Combinatorial Sums Acknowledgements References Chapter 6: On the Exponential Generating Function for Non-backtracking Walks Abstract Motivation Background Exponential Generating Function for Undirected Graphs Exponential Generating Function for Directed Graphs Computing the Centrality Vectors Block Matrix Interpretations Star Graph Analysis Numerical Tests Summary Acknowledgements References Chapter 7: Discrete Approximation ay a Dirichlet Series Connected to the Riemann Zeta-function Abstract Introduction Statement Of The Main Theorem Proof Of Theorem 2 Distance Between (s) and ζuN(s) Proof of Theorem 3 Conclusions Author Contributions References Chapter 8: Dirichlet Series Expansions of P-adic L-functions Abstract Introduction Expansions of P-Adic L-Functions Regularized Bernoulli Distributions Expansions for Different Regularization Parameters Acknowledgements References Section 2 Asymptotic Methods Chapter 9: Asymptotic Analysis of Regular Sequences Abstract Part I: Introduction Synopsis: The Objects of Interest and the Result How to Read this Paper User-friendly Main Result and a First Example Application Overview of the Full Results and Proofs Overview of the Examples Full Results Remarks on the Definitions Part II: Examples Esthetic Numbers Pascal’s Rhombus Part III: Proofs Additional Notations Decomposition into Periodic Fluctuations: Proof of Theorem B Meromorphic Continuation of the Dirichlet Series: Proof of Theorem D Fourier Coefficients: Proof of Theorem E Proof of Theorem A Proof of Proposition 6.4 Part IV: Computational Aspects Strategy for Computing the Fourier Coefficients Details on the Numerical Computation Non-vanishing Coefficients Acknowledgements References Chapter 10: Lagrange Inversion and Combinatorial Species with Uncountable Color Palette Abstract Introduction Preliminaries Main Results Lagrange-good Inversion. Proof of Theorem 3.1 Discussion Appendices Acknowledgements References Chapter 11: On Solving Some Functional Equations Abstract Introduction Lagrange Inversion Equation The Generalized Lagrange Inversion Equation Acknowledgements References Chapter 12: Khinchin Families And Hayman Class Abstract Introduction Khinchin Families Gaussian Khinchin Families Exponentials and Hayman Class Some Examples of Exponentials in the Hayman Class Partitions Acknowledgements References Section 3: Applications to Pure Mathematics and Applied Science Chapter 14: Generating Functions for Generalized Stirling Type Numbers, Array Type Polynomials, Eulerian Type Polynomials and Their Applicat Abstract Introduction, Definitions and Preliminaries Generating Function for Generalized Stirling Type Numbers of the Second Kind Generalized Array Type Polynomials Generalized Eulerian Type Numbers and Polynomials New Identities Involving Families of Polynomials Author’s Contributions Acknowledgements References Chapter 15:Generalized Tepper’s Identity and its Application Abstract Introduction Generalized Tepper’s Identity New Identities Related to the Bernoulli and Euler Polynomials Taylor Series for the Generalized Tepper Identity Conclusions Acknowledgments References Chapter 16: New Bell–sheffer Polynomial Sets Abstract Introduction Sheffer Polynomials New Bell–sheffer Polynomial Sets Iterated Bell–sheffer Polynomial Sets The General Case Conclusions Author Contributions References Index Back Cover