Several $q$-series related to Ramanujan's theta functions

TL;DR

This paper investigates the signs and arithmetic properties of coefficients in specific $q$-series related to Ramanujan's theta functions, providing dissections, identities, and combinatorial interpretations.

Contribution

It introduces new $q$-series identities, 5-dissections, and combinatorial interpretations, expanding understanding of Ramanujan's theta functions and their coefficient behaviors.

Findings

Derived 5-dissections of two $q$-series

Established arithmetic relations for these $q$-series

Presented four new $q$-series identities

Abstract

Quite recently, the first author investigated vanishing coefficients of the arithmetic progressions in several $q$-series expansions. In this paper, we further study the signs of coefficients in two $q$-series expansions and establish some arithmetic relations for several $q$-series expansions by means of Ramanujan's theta functions. We obtain the 5-dissections of these two $q$-series and give combinatorial interpretations for these dissections. Moreover, we obtain four $q$-series identities involving the aforementioned $q$-series, two of which were proved by Kim and Toh via modular forms.