On the computations of contiguous relations for $_{2}F_{1}$ hypergeometric series

Medhat A. Rakha; Adel K. Ibrahim; Arjun K. Rathie

doi:10.4134/CKMS.2009.24.2.291

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2009; 24(2): 291-302

Printed June 1, 2009

https://doi.org/10.4134/CKMS.2009.24.2.291

On the computations of contiguous relations for $_{2}F_{1}$ hypergeometric series

Medhat A. Rakha, Adel K. Ibrahim, and Arjun K. Rathie

Suez Canal University, Suez Canal University, and MIT Engineering College

Abstract

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Abstract

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 $_{2}F_{1}[a_{1},a_{2};a_{3};z]$ with arbitrary integer shifts is presented. Our analysis depends on using shifted operators attached to the three parameters $a_{1},a_{2}$ and $a_{3}$. We also, discussed the existence condition of our formula.

Keywords: hypergeometric function, contiguous relations

MSC numbers: 33C05, 33D15