Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 074, 22 pages      arXiv:1504.08144      https://doi.org/10.3842/SIGMA.2015.074
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

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

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

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.

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

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