A multidimensional generalization of the Barnes integral and the continuous Hahn polynomial
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.
scientific article; zbMATH DE number 1309046
scientific article