Contiguity relations for generalized hypergeometric functions

Abstract:

It is well known that the hypergeometric functions \[ _2{F_1}(\alpha \pm 1,\beta ,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta \pm 1,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta ,\gamma \pm 1;t),\] which are contiguous to $_2{F_1}(\alpha ,\beta ,\gamma ;t)$, can be expressed in terms of \[ _2{F_1}(\alpha ,\beta ,\gamma ;t)\quad {\text {and}}{\quad _2}F_1^\prime (\alpha ,\beta ,\gamma ;t).\] We explain how to derive analogous formulas for generalized hypergeometric functions. Our main point is that such relations can be deduced from the geometry of the cone associated in a recent paper by B. Dwork and F. Loeser to a generalized hypergeometric series.