Class Of Hypergeometric Polynomials Research Articles

Asymptotic distribution of the zeros of a certain family of generalized hypergeometric polynomials

The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: q + 1 F q ( − n , kn + α , … , kn + α + q − 1 q ; kn + β , … , kn + β + q − 1 q ; z ) ( n → ∞ ) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by α < β . By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves.

Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials $$\small {}_{A}\text {F}_{B}\left[ \begin{array}{c} -n,a_2,\ldots ,a_A \\ b_1,b_2,\ldots ,b_B \end{array} ; \begin{array}{cc} z \end{array}\right] $$ , using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it.

Summation identities involving certain classes of polynomials

In some recent investigations involving differential operators for a general family of Lagrange polynomials, Chen et al. [Some new results for the Lagrange polynomials in several variables, ANZIAM J. 49 (2007), pp. 243–258] encountered and proved a certain summation identity for the Chan–Chyan–Srivastava polynomials. With a view to extending and generalizing the aforementioned summation identity, we derive the corresponding results for one class of hypergeometric polynomials, the Jacobi polynomials, the extended Jacobi polynomials, the Laguerre polynomials, the Hermite polynomials, the Lagrange–Hermite polynomials and the Erkuş–Srivastava polynomials.

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

Abstract The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesaro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.

Some double-series identities and associated generating-function relationships

The main object of the present work is to investigate several families of double-series identities and their applications to the Srivastava–Daoust hypergeometric function in two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered.

Series identities and associated families of generating functions

The authors investigate several families of double-series identities as well as their (known or new) consequences involving various hypergeometric functions in one and two variables. A number of associated generating-function relationships, involving certain classes of hypergeometric polynomials, are also considered.

Certain classes of finite-series relationships and generating functions involving the generalized Bessel polynomials

Recently, by making use of the familiar group-theoretic (Lie algebraic) method, a certain mixed trilateral finite-series relationship was proven for the generalized Bessel polynomials. The main object of this paper is to derive several substantially more general families of bilinear, bilateral, and mixed multilateral finite-series relationships and generating functions for these and other related classes of hypergeometric polynomials. A duly corrected and modified analogue of the aforementioned trilateral finite-series relationship is shown to follow by suitably specializing one of the general results presented here. Several closely related (presumably new) finite-series relationships and generating functions, some of which also involve (for example) the Stirling numbers of the second kind, are onsidered rather briefly.

Some families of hypergeometric transformations and generating relations

By applying a quadratic transformation for the Gauss hypergeometric function, the authors derive a family of generating relations for a general polynomial system. Several interesting consequences of the main result, involving various classes of hypergeometric polynomials, are considered. Further generating relations associated with the Laguerre functions and polynomials are also investigated.

Some Clebsch-Gordan type linearisation relations and other polynomial expansions associated with a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications

In this sequel to Srivastava's work (1987), a number of new expansions (in series of various classes of hypergeometric polynomials) are derived for the general multivariable hypergeometric function which provides an interesting and useful unification of numerous families of hypergeometric functions of one, two or more variables encountered naturally in a wide variety of physical and quantum chemical applications. By suitably specialising some of these polynomial expansions, Clebsch-Gordan type linearisation relations are deduced for the products of several Jacobi or Laguerre polynomials. It is also shown how one of the linearisation relations involving Laguerre polynomials would yield the corrected (and modified) versions of a couple of results given by Niukkanen (1985).

A class of hypergeometric polynomials

AbstractThe object of the present paper is first to derive an interesting unification (and generalization) of a fairly large number of finite summation formulas including, for example, those that appeared in this Journal recently. We then briefly remark on its various (known or new) special cases which are associated with certain classes of hypergeometric polynomials in one and two variables. We also give several further generalizations (involving multiple series with essentially arbitrary terms) which are shown to be applicable in the derivation of analogous summation formulas for hypergeometric series (and polynomials) in three and more variables. Finally, with a view to presenting relevance of these types of results in various seemingly diverse areas of applied sciences and engineering, some indication of applicability is provided.

A class of hypergeometric polynomials

In this paper we study the properties of the polynomials $$_3 F_2 \left[ {\begin{array}{*{20}c} { - n,n + \gamma + 1,\zeta ;x} {1 + \alpha ,1 + \beta } \end{array} } \right]$$ .

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order, II

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order, II

Basic series corresponding to a class of hypergeometric polynomials

Letgn(z) be a polynomial of degreeninz. Then there exist constants Πk, nsuch that.

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order

Asymptotic expansions of a class of hypergeometric polynomials with respect to the order