Asymptotic expansions of the hypergeometric function with two large parameters $-$ application to the partition function of a lattice gas in a field of traps

TL;DR

This paper develops new asymptotic expansions for the Gauss hypergeometric function with two large parameters, valid across the entire complex plane, enabling better analysis of physical models like lattice gases in trap fields.

Contribution

It introduces comprehensive asymptotic expansions for ${}_2F_1(a,b;c;z)$ with two large parameters, overcoming previous domain limitations and applicable near singularities.

Findings

Effective asymptotic expansions near singularities

Accurate approximation of the partition function in various limits

Enhanced understanding of hypergeometric functions in physical models

Abstract

The canonical partition function of a two-dimensional lattice gas in a field of randomly placed traps, like many other problems in physics, evaluates to the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$ in the limit when one or more of its parameters become large. This limit is difficult to compute from first principles, and finding the asymptotic expansions of the hypergeometric function is therefore an important task. While some possible cases of the asymptotic expansions of ${}_2F_1(a,b;c;z)$ have been provided in the literature, they are all limited by a narrow domain of validity, either in the complex plane of the variable or in the parameter space. Overcoming this restriction, we provide new asymptotic expansions for the hypergeometric function with two large parameters, which are valid for the entire complex plane of $z$ except for a few specific points. We show that these expansions work well even when we approach the possible singularity of ${}_2F_1(a,b;c;z)$, $|z|=1$, where the current expansions typically fail. Using our results we determine asymptotically the partition function of a lattice gas in a field of traps in the different possible physical limits of few/many particles and few/many traps, illustrating the applicability of the derived asymptotic expansions of the HGF in physics.