Integral representations of the Wilson polynomials and the continuous dual Hahn polynomials
type: Item
instance of: scholarly article
title: Integral representations of the Wilson polynomials and the continuous dual Hahn polynomials
zbMATH Open document ID: 0942.33005
author: Katsuhisa Mimachi
published in: Advances in Applied Mathematics
publication date: 2000-08-21T00:00:00Z
review text: Let \(\{p_n(x)\}_{n=0,1, \dots}\) denote the set of monic orthogonal polynomials, \(p_n(x)\) of degree \(n\), associated with the weight function \(w(x)\). Then it is straightforward to show \[ p_n(x)={1\over C} \int^\infty_{-\infty} dx_1w(x_1) \dots\int^\infty_{- \infty} dx_Nw(x_N) \prod^N_{l=1} (x-x_1) \prod_{1\leq j<k\leq N} (x_k-x_j)^2\tag{1} \] with \(C= \prod^{N-1}_{j=0} \int^\infty_{-\infty} w(x)(p_j(x))^2dx\). With the factor \(\prod_{1\leq j<k\leq N} (x_k-x_j)^2\) replaced by \[ \prod_{1\leq j<k \leq n}(x_k-x_j)^{2c}, \] and a suitable \(C\), it was shown by \textit{K. Aomoto} [SIAM J. Math. Anal. 18, 545-549 (1987; Zbl 0639.33001)] that (1) remains valid for the Jacobi weight \(w(x)= x^{\lambda_1} (1-x)^{\lambda_2}\), \(0<x<1\), \(w(x)=0\) otherwise. Since then these so-called Selberg correlation integrals have been the subject of much study and are now known to be intimately related to the Jack and Macdonald multivariable polynomials. In the present work an analogue of (1) is given for the Wilson and continuous Hahn polynomials, with the integrand a multidimensional generalization of Barnes type integrals. It remains to extend these results to the multidimensional setting.
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1394762
zbMATH Keywords: Wilson polynomials
zbMATH Keywords: continuous Hahn polynomials
reviewed by: Peter J. Forrester
MaRDI profile type: Publication
full work available at URL: aama.1999.0655
OpenAlex ID: W2035793268
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DOI: 10.1006/AAMA.1999.0655
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description: scientific article; zbMATH DE number 1394762