On the computations of contiguous relations for ₂F₁ hypergeometric series

Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a general formula joining three Gauss functions of the form ₂F₁[a₁, a₂; a₃; z] with arbitrary integer shifts is presented. Our analysis depends on using shifted operators attached to the three parameters a₁, a₂ and a₃. We also, discussed the existence condition of our formula.