Matrix coefficients of the large discrete series representations of Sp(2;R) as hypergeometric series of two variables

Abstract

We investigate the radial part of the matrix coefficients with minimal $K$-types of the large discrete series representations of $Sp(2;\mathbf{R})$. They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.