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

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