The matrix-valued hypergeometric equation

The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11-13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123-162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39-83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127-172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1-24]. The hypergeometric function known also as Gauss' function is the unique solution of the hypergeometric equation analytic at z = 0 and with value 1 at z = 0. This function, because of its remarkable properties, has been used for centuries in the whole subject of special functions. In this article we give a matrix-valued analog of the hypergeometric differential equation and of Gauss' function. One can only speculate that many of the connections that made Gauss' function a vital part of mathematics at the end of the 20th century will be shared by its matrix-valued version, discussed here.