Beta-integrals and finite orthogonal systems of Wilson polynomials

Yu. A. Neretin

Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 1071-1089

Résumé

The integral $$ \frac1{2\pi}\int_{-\infty}^\infty\biggl|\frac{\prod_{k=1}^3\Gamma(a_k+is)} {\Gamma(2is)\Gamma(b+is)}\biggr|^2\,ds =\frac{\Gamma(b-a_1-a_2-a_3)\prod_{1\leqslant k<l\leqslant 3}\Gamma(a_k+a_l)} {\prod_{k=1}^3\Gamma(b-a_k)} $$ 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.

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