Families Of Hypergeometric Polynomials Research Articles

On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle

In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$. Differential equations of orders $\rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.

On some classical type Sobolev orthogonal polynomials

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: 2F2(−n,1;α+1,κ+1;x) and 3F2(−n,n+α+β+1,1;α+1,κ+1;x) (α,β,κ>−1, n=0,1,…), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: Ly+λnDy=0, where L,D are linear differential operators with polynomial coefficients not depending on n. For nonnegative integer values of the parameter κ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained. Some generalizations of these polynomials are discussed as well.

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre () and Gegenbauer () polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as of the associated entropy-like integral functionals of these two families of hypergeometric 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 families of hypergeometric polynomials and associated multiple integral representations

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. 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 families of generating functions for a certain class of three-variable polynomials

The present investigation is a continuation of the works initiated by Srivastava [A contour integral involving Fox's H-function, Indian J. Math. 14 (1972), pp. 1–6] and followed by several papers such as (among others) [A. Altın, E. Erkuş, and M.A. Özarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (2006), pp. 315–320; B. González, J. Matera, and H.M. Srivastava, Some q-generating functions and associated generalized hypergeometric polynomials, Math. Comput. Modelling 34 (2001), pp. 133–175; S.-D. Lin, Y.-S. Chao, and H.M. Srivastava, Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl. 294 (2004), pp. 399–411; S.-D. Lin, H.M. Srivastava, and P.-Y. Wang, Some families of hypergeometric transformations and generating relations, Math. Comput. Modelling 36 (2002), pp. 445–459; E. Özergin, M.A. Özarslan, and H.M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Math. Comput. Modelling 50 (2009), pp. 1113–1120]. In this study, we introduce a family of three-variable polynomials and derive a number of two-sided linear generating functions between these polynomials and another family of two-variable polynomials. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials. Several other results of the above-mentioned types are also considered.