New representations for $\sigma(q)$ via reciprocity theorems

TL;DR

This paper introduces two novel representations for Ramanujan's sigma function using advanced reciprocity theorems, involving free complex parameters, which could enhance analytical and computational approaches.

Contribution

It presents new parameterized representations of sigma(q) based on three- and four-variable reciprocity theorems, expanding the toolkit for studying Ramanujan's functions.

Findings

Two new parameterized representations of sigma(q)

Use of three-variable reciprocity theorem for the first representation

Use of four-variable reciprocity theorem for the second representation

Abstract

Two new representations for Ramanujan's function $\sigma(q)$ are obtained. The proof of the first one uses the three-variable reciprocity theorem due to Soon-Yi Kang and a transformation due to R.P. Agarwal while that of the second uses the four-variable reciprocity theorem due to George Andrews and a generalization of a recent transformation of Andrews, Schultz, Yee and the second author. The advantage of these representations is that they involve free complex parameters - one in the first representation, and two in the second.