Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials

TL;DR

The paper studies the asymptotics of a family of Sobolev orthogonal polynomials defined by a $\Delta$-Sobolev inner product $\langle p,q\rangle = L[pq] + \lambda L[\Delta p\,\Delta q]$ with generalized Charlier weights. It develops a comprehensive asymptotic framework by relating the Sobolev polynomials $S_n(x;\lambda,z)$ to the non-Sobolev polynomials $P_n(x;z)$, and derives explicit large-$n$ expansions in terms of the falling factorial basis $\varphi_n(x)$, including the leading norms and coefficient recurrences. The main result provides an explicit expansion $\displaystyle \frac{S_n(x;\lambda,z)}{\varphi_n(x)} \sim \sum_{k\ge0} \sigma_k(x;z)\, n^{-k}$ with coefficients depending on $x$, $z$, $b$, and $\lambda$, up to $\sigma_4$. These contributions illuminate the structure of $\Delta$-Sobolev orthogonality for discrete semiclassical weights and offer practical asymptotics for computation and further analysis.

Abstract

In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ--Sobolev orthogonality. We obtain an asymptotic expansion for this orthogonal polynomials where the falling factorial polynomials play an important role.

Theorems & Definitions (15)

• Remark 1
• Lemma 2
• Proposition 3
• Remark 4
• Proposition 5
• Lemma 6
• Corollary 7
• Theorem 8
• Remark 9
• Theorem 10