A multidimensional generalization of the Barnes integral and the continuous Hahn polynomial
type: Item
instance of: scholarly article
title: A multidimensional generalization of the Barnes integral and the continuous Hahn polynomial
zbMATH Open document ID: 0933.33017
author: Katsuhisa Mimachi
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-03-27T00:00:00Z
review text: In [J. Phys. A, Math. Gen. 28, L369--L374 (1995; Zbl 0859.33019)], \textit{J. F. van Diejen} considered multivariable continuous Hahn polynomials. The orthogonality relation of these polynomials is based on a multidimensional generalization of the Barnes integral. Let the corresponding integral kernel be \(\Phi(t_1,\dots,t_n)\). In the present paper it is shown that the integral \[ \int _{-\infty} ^{\infty}\cdots\int _{-\infty} ^{\infty} \Phi(t_1,\dots,t_n)\prod _{k=1} ^{n}(t_k-z) dt_1\dots dt_n \] is, up to a multiplicative constant, equal to a continuous Hahn polynomial (in \(z\)). The proof proceeds by demonstrating that the integral satisfies the same difference equation (in \(z\)) as the continuous Hahn polynomials.
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 1309046
zbMATH Keywords: multidimensional Barnes integral
zbMATH Keywords: continuous Hahn polynomials
zbMATH Keywords: van Diejen
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
OpenAlex ID: W2009266538
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DOI: 10.1006/JMAA.1999.6323
description: scientific article; zbMATH DE number 1309046