On singular univariate specializations of bivariate hypergeometric functions

TL;DR

This paper explores the complexities and potential pitfalls in evaluating univariate specializations of Appell's hypergeometric functions, highlighting issues with naive approaches and illustrating them through concrete examples.

Contribution

It provides a detailed analysis of the divergence and branching issues in univariate specializations of bivariate hypergeometric functions, emphasizing the need for careful evaluation methods.

Findings

Naive evaluation at x=0 can lead to divergence errors.

Univariate specializations may involve branching terms.

Concrete examples demonstrate potential evaluation pitfalls.

Abstract

It is tempting to evaluate F2(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for 2F1(1). But this kind of naive evaluation can lead to errors as the 2F1(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications on concrete examples.