Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials

Abstract

This is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations. ᮊ

Asymptotic analysis of the Hermite polynomials from their differential–difference equation

“…We will consider the case x ≈ √ 2n and find the corresponding result for x ≈ − √ 2n by using (2). From (24) we have…”

“…Being the limiting case of several families of classical orthogonal polynomials [16], they are of fundamental importance in asymptotic analysis [24], [33]. The Hermite polynomials satisfy the orthogonality condition…”

A generalization of Hermite polynomials

“…We refer some of them beginning with [1], followed by [4], [7], [9] and [5]. We also refer some recent work regarding properties, extensions, generalizations and applications of Hermite and the related orthogonal polynomials; e.g.…”

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

“…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.…”