Transformation formulas of a character analogue of $\log\theta_{2}(z)$

TL;DR

This paper derives transformation formulas for a character analogue of a logarithmic theta function, generalizing Hardy--Berndt sums, and applies these to establish reciprocity formulas and series relations.

Contribution

It introduces new transformation formulas for a character analogue of \\log\\theta_{2}(z), generalizing Hardy--Berndt sums, and derives related reciprocity and series relations.

Findings

Derived transformation formulas for the function involving characters.

Generalized Hardy--Berndt sums and established their reciprocity.

Presented new series relations based on these transformations.

Abstract

In this paper, transformation formulas for the function \[ A_{1}\left(z,s:\chi\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}\chi\left(n\right)\chi\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2\pi imnz/k} \] are obtained. Sums that appear in transformation formulas are generalizations of the Hardy--Berndt sums $s_{j}(d,c),$ $j=1,2,5$. As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented.