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

TL;DR

This paper investigates the asymptotic behavior of biorthogonal polynomial systems related to Hermite and Laguerre polynomials, establishing connections with various classical orthogonal polynomials and verifying the Askey scheme.

Contribution

It introduces a new framework for biorthogonal polynomials approximating Hermite and Laguerre polynomials, deriving their asymptotic relations and applications to classical polynomial families.

Findings

Derived asymptotic relations between orthogonal and combinatorial polynomials.

Established asymptotic representations for several classical polynomials.

Verified the Askey scheme of hypergeometric orthogonal polynomials.

Abstract

In this paper, the structures to a family of biorthogonal polynomials that approximate to the Hermite and Generalized Laguerre polynomials are discussed respectively. Therefore, the asymptotic relation between several orthogonal polynomials and combinatorial polynomials are derived from the systems, which in turn verify the Askey scheme of hypergeometric orthogonal polynomials. As the applications of these properties, the asymptotic representations of the generalized Buchholz, Laguerre, Ultraspherical(Gegenbauer), Bernoulli, Euler, Meixner and Meixner-Pllaczekare polynomials are derived from the theorems directly. The relationship between Bernoulli and Euler polynomials are shown as a special case of the characterization theorem of the Appell sequence generated by $\alpha$ scaling functions.