New results for the series ${}_2F_1(x)$ with an application
Commun. Korean Math. Soc. 2014 Vol. 29, No. 1, 65-74
https://doi.org/10.4134/CKMS.2014.29.1.65
Printed January 31, 2014
Junesang Choi and Arjun Kumar Rathie
Dongguk University, Riverside Transit Campus
Abstract : The well known quadratic transformation formula due to Gauss: $$ (1-x)^{-2a}\, _2F_1 \left[ \aligned a,\, b \, &;\\ 2b \, &; \endaligned \,\, -\frac{4x}{(1-x)^2} \right] ={}_2F_1 \left[ \aligned a,\,a- b+\frac{1}{2} \, &;\\ b+\frac{1}{2} \,&; \endaligned \,\, x^2 \right] $$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series ${}_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.
Keywords : Gamma function, hypergeometric function, generalized hypergeometric function, Gauss's quadratic transformation formula for ${}_2F_1$, Watson's summation theorem for ${}_3F_2(1)$
MSC numbers : Primary 33B20, 33C20; Secondary 33B15, 33C05
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