Barnes-type integral and the Meixner-Pollaczek polynomials
type: Item
instance of: scholarly article
title: Barnes-type integral and the Meixner-Pollaczek polynomials
zbMATH Open document ID: 0935.33010
DOI: 10.1023/A:1007649813978
author: Katsuhisa Mimachi
published in: Letters in Mathematical Physics
publication date: 2000-05-04T00:00:00Z
review text: The author develops an integral representation of Meixner-Pollaczek polynomials \(P_n^{(\alpha/\lambda)}\) in terms of a multidimensional generalization of the Barnes-type integral. The main theorem is \[ \langle f\rangle/\langle 1\rangle= \Biggl[ \prod_{j=1}^n (2\alpha+ (j-1)\lambda)/ (e^{2i\varphi}- 1)^n \Biggr] P_n^{(\alpha/\lambda)} (z/\lambda; \varphi), \] \(\alpha,\lambda>0\), \(0< \varphi< \pi\), where \(\langle f\rangle:= \int_{-i\infty}^{i\infty} \dots \int_{-i\infty}^{i\infty} f(x,t)\Phi(x) dx_1\dots dx_n\) and \(f(x,t):= \prod_{j=1}^n (x_-iz)\). The function \[ \Phi(x):= \prod_{j=1}^n \{e^{(2\varphi-\pi) x_j/i} \Gamma(\alpha+x_j) \Gamma(\alpha-x_j)\} \prod_{1\leq j\neq k\leq n} \{\Gamma(\lambda+ x_j- x_k)/ \Gamma(x_j-x_k)\} \] respects the permutation \(x_j\mapsto x_k\). The work is motivated by the study of Barnes-types integrals from the perspective of the finite-difference version of the de Rham theory. The author lists references where he gives generalizations. An overview corresponding to the Askey scheme is to be given in another paper.
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1347230
zbMATH Keywords: Meixner-Pollaczek polynomials
zbMATH Keywords: Barnes-type integral
zbMATH Keywords: de Rham theory
reviewed by: Peter A. McCoy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1347230