Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers

Alexander Dyachenko and Dmitrii Karp ()
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Alexander Dyachenko: Keldysh Institute of Applied Mathematics, 125047 Moscow, Russia
Dmitrii Karp: Department of Mathematics, Holon Institute of Technology, Holon 5810201, Israel

Mathematics, 2022, vol. 10, issue 20, 1-26

Abstract: Given real parameters a , b , c and integer shifts n 1 , n 2 , m , we consider the ratio R ( z ) = 2 F 1 ( a + n 1 , b + n 2 ; c + m ; z ) / 2 F 1 ( a , b ; c ; z ) of the Gauss hypergeometric functions. We find a formula for Im R ( x ± i 0 ) with x > 1 in terms of real hypergeometric polynomial P , beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples.

Keywords: gauss hypergeometric function; gauss continued fraction; integral representation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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