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

Abstract

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61-85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131-146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623-633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner-Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.

Biorthogonal polynomial systems associate with Hermitepolynomials and generalized Laguerre polynomials

“…. 其中 Askey 格式的底层揭示了一大 类正交多项式: Gegenbauer、Laguerre、Charlier、Jacobi、Meixner-Pollaczek、Meixner、Krawtchouk 和 Hahn 型 (参见文献 [5][6][7][8][9]) 之间的渐近关系, 并且这些渐近形式皆由 Hermite 多项式或者 Laguerre 多项 式所表示. 因而研究渐近于 Hermite 多项式和 Laguerre 多项式的函数列的判定性质显得尤为重要.…”

“…第 3 节给出与广义 Laguerre 多项式相关联的双正交多项式 系统的理论框架, 从而验证了 Askey 格式中若干类超几何多项式的渐近关系. 虽然本文中的若干超 几何多项式渐近性质在文献 [9,10] 中有所讨论, 但是本文利用生成函数的逼近性质, 构造渐近于广义 Laguerre 多项式的函数列, 从而得到渐近于 Laguerre 多项式并且给出具有双正交性质的多项式系统 的判定定理. 本文给出的理论框架是研究正交多项式渐近性质的全新视角, 利用生成函数的渐近性质, 系统地给出了几乎所有 Askey 格式中底层多项式的渐近关系, 因而有别于文献 [9,10] 中针对个别多项 式渐近性质的逐一验证.…”

Zeros of Meixner and Krawtchouk polynomials

“…The zero asymptotics of normalised Krawtchouk polynomials when the ratio of parameter n/N → α as n, N → ∞ was investigated in [13] and [14] by finding the support and density of the constrained extremal measure for all possible values of the parameter α and the asymptotic zero distribution of Meixner polynomials has also been studied by various authors (cf. [17] and [22]).…”

Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials

“…A rich source of orthogonal polynomials, for instance, Gegenbauer [18], Laguerre [19], Charlier [13], Jacobi [14], Meixner-Pollaczek, Meixner, Krawtchouk and Hahn-type polynomials [5] have asymptotic approximations in terms of Hermite polynomials, which is known as the famouse Askey scheme [21,25,20].…”