Contiguous relations and their computations for 2F1 hypergeometric series

The hypergeometric function F12[a"1,a"2;a"3;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 a"1,a"2 and a"3. More precisely, we obtained a recurrence relation including F12[a"[email protected]"1,a"2;a"3;z],F12[a"1,a"[email protected]"2;a"3;z]andF12[a"1,a"2;a"[email protected]"3;z] for any arbitrary integers @a"1,@a"2 and @a"3.