Contiguous relations and their computations for ₂F₁ hypergeometric series

The hypergeometric function ₂F₁ [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, a₂ and a₃. More precisely, we obtained a recurrence relation including ₂F₁ [a₁ + α₁, a₂ ; a₃ ; z], ₂F₁ [a₁, a₂ + α₂ ; a₃ ; z] and ₂F₁ [a₁, a₂ ; a₃ + α₃ ; z] for any arbitrary integers α₁, α₂ and α₃.