Transformation formulas of a character analogue of $\log {\theta }_2(z)$

In this paper, transformation formulas for the function $$A_1(z,s:\chi )=\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}\chi \left(n\right)\chi \left(m\right){(-1)}^m{n}^{s-1}{e}^{2\pi imnz\mathrm{/}k}$$ 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.