Contiguous relations and their computations for F12 hypergeometric series

The hypergeometric function F12[a1,a2;a3;z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a1,a2a1,a2 and a3a3. More precisely, we obtained a recurrence relation including F12[a1+α1,a2;a3;z],F12[a1,a2+α2;a3;z]andF12[a1,a2;a3+α3;z] for any arbitrary integers α1,α2α1,α2 and α3α3.