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

TL;DR

This paper develops fractional integral and generalized Stieltjes transform formulas for hypergeometric functions, revealing their transmutation properties and deriving new integral representations for solutions of the hypergeometric differential equation.

Contribution

It introduces new fractional integration and Stieltjes transform formulas that demonstrate transmutation properties for hypergeometric functions, expanding the analytical tools available for these functions.

Findings

Eight fractional integration formulas derived for hypergeometric functions.

Four generalized Stieltjes transform formulas established for hypergeometric solutions.

New Euler type integral representations obtained for hypergeometric solutions.

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.