Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

Abstract

It has been shown in Ferreira et al. (Adv. ) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

“…Higher coefficients c k , k 3 can be obtained recurrently from a differential equation satisfied by f (x, w) or directly from their definition (3) (using computer algebra programs like Mathematica, Matlab, etc.). In our previous works about polynomials located in the first levels of the Askey-tableau [1,2], we have given recurrent formulas for c k using a differential equation satisfied by f (x, w). However, the functions f (x, w) involved in this paper are more complicated and analytic formulas for c k are too cumbersome to be written down here.…”

“…The limit (1) is obtained from the first order approximation of this expansion. The asymptotic method from which expansions like (2) are obtained was introduced and developed in [1,[7][8][9]. The method to approximate orthogonal polynomials in terms of Hermite, Laguerre and Charlier polynomials is described in [7,9,1], respectively.…”

“…Asymptotic expansions of Meixner-Pollaczek, Jacobi, Meixner and Krawtchouk polynomials in terms of Hermite and Charlier polynomials are given in [1]. Asymptotic expansions of the four Hahn polynomials in terms of Hermite, Laguerre and Charlier polynomials are given in [2].…”

“…Another different method, based on a representation of the Hahn polynomials in terms of a difference operator is considered in [11], in which an asymptotic formula for the Hahn polynomials Q n (x; , , N) in terms of Jacobi polynomials is derived for large values of N. In our method we keep the degree n fixed and let some parameter(s) of the polynomial go to infinity. The purpose of this paper is the continuation of the asymptotic program developed in [1,2,[7][8][9] and derive asymptotic expansions (and limits when it is possible) between the fourth and third levels of the Askey tableau.…”

An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences

“…Other papers include [Al-Salam and Verma 1970;Di Bucchianico 1994;Di Nardo et al 2011;Dominici 2007;Hofbauer 1981;Popa 1997;1998;Shukla and Rapeli 2011]. In addition, a very large amount of work has been completed pertaining to the theory and applications of specific A-type 0 orthogonal sets, e.g., [Akleylek et al 2010;Chen et al 2011;Coffey 2011;Coulembier et al 2011;Dueñas and Marcellán 2011;Ferreira et al 2008;Hutník 2011;Khan et al 2011;Kuznetsov 2008;Miki et al 2011;Mouayn 2010;Sheffer 1941;Vignat 2011;Yalçinbaş et al 2011]. Indeed, research on the Sheffer sequences is an active area and important in its own right.…”