New results on the associated Meixner, Charlier, and Krawtchouk polynomials
type: Item
instance of: scholarly article
title: New results on the associated Meixner, Charlier, and Krawtchouk polynomials
zbMATH Open document ID: 1542.4203
DOI: 10.1080/10652469.2024.2322078
author: Khalid Ahbli
published in: Integral Transforms and Special Functions
publication date: 2024-06-25T00:00:00Z
review text: This paper delves into the study of associated orthogonal polynomials (AOPs), focusing particularly on the associated Meixner polynomials (AMPs), denoted as \(\mathscr{M}_n(x ; \beta, c, \gamma)\). AOPs are defined in terms of a positive measure on the real line and satisfy a three-term recurrence relation. The research extends previous work by establishing explicit expressions and generating functions for AMPs and their connections to other classical polynomials.\N\NLet \(\mathrm{d} \mu\) be a positive measure with finite moments of all orders and an infinite number of points of increase. A family of polynomials \(\{P_n(x)\}_{n \geq 0}\) is orthogonal with respect to \(\mathrm{d} \mu\) if it satisfies: \N\[\N\int_{\mathbb{R}} P_n(x) P_m(x) \mathrm{d} \mu(x) = h_n \delta_{nm}, \quad h_n > 0. \N\]\NOrthogonality requires the polynomials to obey a three-term recurrence relation with real coefficients \(A_n\), \(B_n\), \(C_n\), \(D_n\) such that: \N\[\NA_{n-1} B_{n-1} B_n D_n > 0, \quad n \geq 1. \N\]\NAOPs \(\{P_n(x ; \gamma)\}_{n \geq 0}\) are defined similarly with parameters \(A_{n+\gamma}, B_{n+\gamma}, C_{n+\gamma}, D_{n+\gamma}\), where \(\gamma \geq 0\). Previous studies have explored explicit representations, generating functions, and orthogonality measures for AOPs, particularly those at the top of the Askey and \(q\)-Askey tableaux, including Wilson, Racah, Askey-Wilson, and \(q\)-Racah polynomials.\N\NThis paper focuses on AMPs \(\mathscr{M}_n(x ; \beta, c, \gamma)\), with coefficients: \N\[\NA_{n+\gamma} = c, \quad B_{n+\gamma} = c-1, \quad C_{n+\gamma} = (c+1)(n+\gamma) + \beta c, \quad D_{n+\gamma} = (n+\gamma)(n+\gamma+\beta-1). \N\]\NFollowing Ismail et al.'s work [\textit{M. E. H. Ismail} et al., J. Approx. Theory 55, No. 3, 337--348 (1988; Zbl 0656.60092)], the author presents new explicit expressions for AMPs using connections to associated Meixner-Pollaczek polynomials, deriving a novel generating function and exploring the implications for other p…
reviewed by: Pierluigi Vellucci
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 7870885
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7870885