Character analogues of certain Hardy-Berndt sums

In this paper we consider transformation formulas for $B\left( z,s:χ\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0} ^{\infty}χ(m)χ(2n+1)\left( 2n+1\right) ^{s-1}e^{πim(2n+1)z/k}.$ 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 $s_{3,p}\left( d,c:χ\right)$ and $s_{4,p}\left( d,c:χ\right)$.

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