We investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; 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.

We study a quadruple sequence and express its common limit by Lauricella’s hypergeometric function FD(¼,¼,¼.¼, 1; z1, z2, z3)of three variables. We give a functional equation of FD, which is the key to get our expression of the common limit.

Explicit evaluations of the symmetric Euler integral $\int _{0}^{1}\,{{u}^{\alpha }}{{(1-u)}^{\alpha }}f(u)\,du$ are obtained for some particular functions $f$. These evaluations are related to duplication formulae for Appell’s hypergeometric function ${{F}_{1}}$ which give reductions of ${{F}_{1}}(\alpha ,\beta ,\beta ,2\alpha ,y,z)$ in terms of more elementary functions for arbitrary $\beta $ with $z=y/(y-1)$ and for $\beta =\alpha +\frac{1}{2}$ with arbitrary $y,z$. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at $n$ independent randomtimes with uniformdistribution on $[0,1]$, then the broken line approximation to the bridge obtained from these $n+2$ values has a total variation whose mean square is $n(n+1)/(2n+1)$.

For Lauricella's hypergeometric function F(n)D of n variables, we prove two formulas exhibiting its behaviour near the boundaries of the n-dimensional region of convergence of the multiple series defining it. Each of these results can be applied to deduce the corresponding properties of several simpler hypergeometric functions of one, two, and more variables.