Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions

Abstract

In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.

Higher Transcendental Functions and Their Multi-Disciplinary Applications

“…The geographical distribution of the contributors to this Special Issue is remarkably widely-scattered. Their contributions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) originated in many different countries on every continent of the world.…”

“…Reference [17] happens to be an invited survey-cum-expository review article which contains a brief and comprehensive account of some general families of linear and bilinear generating functions associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results, as well as the methods and techniques used for their derivations, which are presented in [17], are intended to provide incentive and motivation for further research on the subject reviewed in [17].…”

“…It may be of interest to remark in conclusion that the above-mentioned work [17] is, in some sense, motivated by an earlier survey-cum-expository review article [18] which presented a brief introductory overview and survey of some of the recent developments in the theory of several extensively-studied higher transcendental functions and their potential applications. It is in [18] in which the interested reader can find some developments involving a hybrid version of several known extensions and generalizations of the Mittag-Leffler type functions, as well as the Hurwitz-Lerch type zeta functions, together with its associated fractional integrals and fractional derivatives (see, for details, [19,20]; see also [21]).…”

Laguerre-Type Bernoulli and Euler Numbers and Related Fractional Polynomials

“…The consideration of generalized forms of polynomials and classical numbers has received much attention in recent years, as can be seen in the [1][2][3][4][5]. This has often been achieved by exploiting the generating functions [6][7][8].…”

Some generating functions of the Bessel and related orthogonal polynomials

“…The motivation and interest for the study of various families of generating functions lie in their role in the investigation of various potentially useful properties and characteristics of the sequences which they generate. In the form of Z-transforms, which essentially are the discrete counterparts of the Laplace transform, generating functions are used in converting difference equations of discretetime signals and systems into algebraic equations, thereby providing simplifications in discrete-time system analysis, and also in a wide variety of problems involving sequential fractional-order difference operators, operations research and other areas of applied sciences (including, for example, queuing theory and related stochastic processes) (see, for details, [36] and the references which are cited therein).…”