Contiguous relations for 2F1 hypergeometric series

Abstract. Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous rela-tions 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. 1