  In this paper we consider transformation formulas for \[ B\left(
z,s:\chi\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0}
^{\infty}\chi(m)\chi(2n+1)\left( 2n+1\right) ^{s-1}e^{\pi im(2n+1)z/k}. \] We
derive reciprocity theorems for the sums arising in these transformation
formulas and investigate certain properties of them. With the help of the
character analogues of the Euler--Maclaurin summation formula we establish
integral representations for the Hardy-Berndt character sums $s_{3,p}\left(
d,c:\chi\right) $ and $s_{4,p}\left( d,c:\chi\right) $.
