  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.
