Barnes-type integral and the Meixner-Pollaczek polynomials
The author develops an integral representation of Meixner-Pollaczek polynomials \(P_n^{(\alpha/\lambda)}\) in terms of a multidimensional generalization of the Barnes-type integral. The main theorem is \[ \langle f\rangle/\langle 1\rangle= \Biggl[ \prod_{j=1}^n (2\alpha+ (j-1)\lambda)/ (e^{2i\varphi}- 1)^n \Biggr] P_n^{(\alpha/\lambda)} (z/\lambda; \varphi), \] \(\alpha,\lambda>0\), \(0< \varphi< \pi\), where \(\langle f\rangle:= \int_{-i\infty}^{i\infty} \dots \int_{-i\infty}^{i\infty} f(x,t)\Phi(x) dx_1\dots dx_n\) and \(f(x,t):= \prod_{j=1}^n (x_-iz)\). The function \[ \Phi(x):= \prod_{j=1}^n \{e^{(2\varphi-\pi) x_j/i} \Gamma(\alpha+x_j) \Gamma(\alpha-x_j)\} \prod_{1\leq j\neq k\leq n} \{\Gamma(\lambda+ x_j- x_k)/ \Gamma(x_j-x_k)\} \] respects the permutation \(x_j\mapsto x_k\). The work is motivated by the study of Barnes-types integrals from the perspective of the finite-difference version of the de Rham theory. The author lists references where he gives generalizations. An overview corresponding to the Askey scheme is to be given in another paper.
scientific article; zbMATH DE number 1347230