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

Abstract

In this paper, transformation formulas for the function \[ A_{1}\left(z,s:χ\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}χ\left(n\right)χ\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2πimnz/k} \] 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.

References in corpus (3)

• Character analogues of certain Hardy-Berndt sums
• Character analogue of the Boole summation formula with applications
• Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

Cited by in corpus (1)

• Reciprocity formulas for Hall-Wilson-Zagier type Hardy-Berndt sums