A naive-topological study of the contiguity relations for hypergeometric functions

Masaaki Yoshida
2005 PDEs, Submanifolds and Affine Differential Geometry   unpublished
When the parameters are real, the hypergeometric equation defines a Schwarz triangle. We study a combinatorial-topological property of the Schwarz triangle when the three angles are not necessarily less than π. 1. Introduction. Let E(a, b, c) be the hypergeometric differential equation where a, b and c are real parameters. Its Schwarz map (cf. [Yo]) is defined by where u 1 and u 2 are two linearly independent solutions of E(a, b, c). The image of the upper half plane then the Schwarz triangle
more » ... Schwarz triangle is indeed a triangle bounded by three arcs with angles |1 − c|π, |c − a − b|π and |b − a|π, in this order. But otherwise, the Schwarz triangle T may be fairly complicated and it would cover the sphere Z many times. In this paper, we study the Schwarz triangle T in a naive topological manner. It is a bit surprising that no one has ever studied this fundamental problem seriously. As the reader will see, the result is indeed not so simple. To formulate our problem, we need to fix some notation. In any case, as far as the parameters are real, the images of the intervals (−∞, 0), (0, 1) and (1, +∞) under the restriction s| X + of s are (parts of) circles; we call these circles C (−∞,0) , C (0,1) and C (1,+∞) , respectively. If x tends to 0 in X + then s(x) converges; we 2000 Mathematics Subject Classification: Primary 33C05.
doi:10.4064/bc69-0-20 fatcat:eh6fkdlntfaf3gu3s4fyolowoe