First memoir on the asymptotics of certain infinite products

TL;DR

This paper explores the asymptotic behavior of certain infinite products related to Rogers-Ramanujan identities, proposing a new approach to their proof via asymptotics of $q$-identities and finite versions.

Contribution

It introduces the first detailed analysis of the asymptotics of these infinite products, offering a novel perspective for proving Rogers-Ramanujan type identities.

Findings

Asymptotic formulas for specific infinite products derived.

Potential application of asymptotics in proving modular identities.

Connections between finite $q$-identities and their infinite counterparts.

Abstract

The product sides of the Rogers--Ramanujan identities and alike often appear to be "transparently modular" (functions). The old work by Rogers (1894) and recent work by Rosengren make use (somewhat implicitly) of this fact for proving the identities with the help of underlying modular equations$-$the main challenge is verifying the latter for the sum sides. Here we speculate on the potentials of using the asymptotics of such $q$-identities or their finite versions for proving them.