Identities between Appell's and hypergeometric functions

TL;DR

This paper classifies special cases where Appell's hypergeometric functions reduce to simpler hypergeometric functions, revealing identities and transformations that connect these functions.

Contribution

It provides a classification of univariate specializations of Appell's functions that satisfy second-order Fuchsian equations and establishes explicit relations with hypergeometric functions.

Findings

Identified cases where Appell's functions reduce to hypergeometric functions

Derived explicit identities and transformations between these functions

Discussed computational aspects of these identities

Abstract

Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2, and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.