Variations on hypergeometric functions

TL;DR

This paper develops new integral formulas for hypergeometric functions, studies their asymptotic expansions, and explores their variations and related differential equations, with applications to multiple zeta values and WKB analysis.

Contribution

It introduces novel integral formulas, analyzes hypergeometric variations, and clarifies the behavior of related differential equations, including corrections to previous results.

Findings

New integral formulas for hypergeometric functions.

Analysis of WKB expansions and variations for small perturbations.

Correction of a previous claim about the WKB expansion of elta_3(mbda).

Abstract

We prove new integral formulas for generalized hypergeometric functions and their confuent variants. We apply them, via stationary phase formula, to study WKB expansions of solutions: for large argument in the confuent case and for large parameter in the general case. We also study variations of hypergeometric functions for small perturbations of hypergeometric equations, i.e., in expansions of solutions in powers of a small parameter. Next, we present a new proof of a theorem due to Wasow about equivalence of the Airy equation with its perturbation; in particular, we explain that this result does not deal with the WKB solutions and the Stokes phenomenon. Finally, we study hypergeometric equations, one of second order and another of third order, which are related with two generating functions for MZVs, one $\Delta_2 (\lambda )$ for $\zeta(2, \ldots , 2)$'s and another $\Delta_3 (\lambda )$ for $\zeta(3, \ldots , 3)$'s; in particular, we correct a statement from [ZZ3] that the function $\Delta_3(\lambda)$ admits a regular WKB expansion.