Integral representations of the Wilson polynomials and the continuous dual Hahn polynomials
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.
scientific article; zbMATH DE number 1394762
scientific article