This article is cited in 8 papers

Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators

Tom H. Koornwinder

Korteweg-de Vries Institute, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Abstract: For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on the solutions. By specialization two Euler type integral representations for each of the six solutions are obtained.

Keywords: Gauss hypergeometric function; Euler integral representation; fractional integral transform; Stieltjes transform; transmutation formula.

MSC: 33C05; 44A15; 44A20; 26A33

Received: April 29, 2015; in final form September 14, 2015; Published online September 20, 2015

Language: English

DOI: 10.3842/SIGMA.2015.074

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ArXiv: 1504.08144