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

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.