Contiguous relations and their computations for hypergeometric series

The hypergeometric function 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,a2 and a3. More precisely, we obtained a recurrence relation including for any arbitrary integers α1,α2 and α3.