New results on the associated Meixner, Charlier, and Krawtchouk polynomials (Q6561456)

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 polynomials like Charlier and Krawtchouk.