Contiguous relations and their computations for F12 hypergeometric series

Authors:
Adel K. Ibrahim;Medhat A. Rakha
Affiliations:
Mathematics Department, College of Science, Suez Canal University, Ismailia 41522, Egypt;Mathematics Department, College of Science, Suez Canal University, Ismailia 41522, Egypt
Venue:
Computers & Mathematics with Applications
Year:
2008

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Abstract

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"1+@a"1,a"2;a"3;z],F12[a"1,a"2+@a"2;a"3;z]andF12[a"1,a"2;a"3+@a"3;z] for any arbitrary integers @a"1,@a"2 and @a"3.