Treffer: Contiguity Relations for Generalized Hypergeometric Functions: Contiguity relations for generalized hypergeometric functions

It is well known that the hypergeometric functions \[ 2 F 1 ( α ± 1 , β , γ ; t ) , 2 F 1 ( α , β ± 1 , γ ; t ) , 2 F 1 ( α , β , γ ± 1 ; t ) , _2{F_1}(\alpha \pm 1,\beta ,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta \pm 1,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta ,\gamma \pm 1;t), \] which are contiguous to 2 F 1 ( α , β , γ ; t ) _2{F_1}(\alpha ,\beta ,\gamma ;t) , can be expressed in terms of \[ 2 F 1 ( α , β , γ ; t ) and 2 F 1 ′ ( α , β , γ ; t ) . _2{F_1}(\alpha ,\beta ,\gamma ;t)\quad {\text {and}}{\quad _2}F_1^\prime (\alpha ,\beta ,\gamma ;t). \] 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.