Beta-integrals and finite orthogonal systems of Wilson polynomials
Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 1071-1089
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The integral
$$
\frac1{2\pi}\int_{-\infty}^\infty\biggl|\frac{\prod_{k=1}^3\Gamma(a_k+is)}
{\Gamma(2is)\Gamma(b+is)}\biggr|^2\,ds
=\frac{\Gamma(b-a_1-a_2-a_3)\prod_{1\leqslant k\leqslant 3}\Gamma(a_k+a_l)}
{\prod_{k=1}^3\Gamma(b-a_k)}
$$
is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a result the system of orthogonal polynomials is finite.
Systems of orthogonal polynomials related to ${}_5H_5$-Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside
the domain of positiveness of the usual weight.
@article{SM_2002_193_7_a5,
author = {Yu. A. Neretin},
title = {Beta-integrals and finite orthogonal systems of {Wilson} polynomials},
journal = {Sbornik. Mathematics},
pages = {1071--1089},
publisher = {mathdoc},
volume = {193},
number = {7},
year = {2002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2002_193_7_a5/}
}
Yu. A. Neretin. Beta-integrals and finite orthogonal systems of Wilson polynomials. Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 1071-1089. http://geodesic.mathdoc.fr/item/SM_2002_193_7_a5/