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

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.

References in corpus (5)

• Transmutations and Applications: a survey
• Hypergeometric type functions and their symmetries
• Buschman-Erdelyi transmutations, classification and applications
• A derivation of two transformation formulas contiguous to that of Kummer's second theorem via a differential equation approach
• A derivation of two quadratic transformations contiguous to that of Gauss via a differential equation approach

Cited by in corpus (4)

• From Conformal Group to Symmetries of Hypergeometric Type Equations
• Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples
• Representations of hypergeometric functions for arbitrary parameter values and their use
• On two classes of generalized fractional operators (with short historical survey of fractional calculus)