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

Author

Listed:
  • Alexander Dyachenko

    (Keldysh Institute of Applied Mathematics, 125047 Moscow, Russia)

  • Dmitrii Karp

    (Department of Mathematics, Holon Institute of Technology, Holon 5810201, Israel
    School of Economics and Management and Far Eastern Center for Research and Education in Mathematics, Far Eastern Federal University, 690922 Vladivostok, Russia)

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.

Suggested Citation

  • Alexander Dyachenko & Dmitrii Karp, 2022. "Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers," Mathematics, MDPI, vol. 10(20), pages 1-26, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:20:p:3903-:d:948473
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    Cited by:

    1. Sergei Sitnik, 2023. "Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”," Mathematics, MDPI, vol. 11(15), pages 1-7, August.

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