Higher-order character Dedekind sum

Abstract

In this paper, we are interested in higher-order character Dedekind sum $sum_{v=0}^{ck;-1};X_1;(nu);{mathfrak B}_{p,;x_2};left(afrac{v;+;z}c+xright);+;{mathfrak B}_q;left(bfrac{v;+;z}{ck}+yright);,;a,;b,;c;in;mathbb{N};and;x,;y,;z;in;mathbb{R}$ where χ1 and χ2 are primitive characters of modulus k, Bp (x) and Bp,χ2(x) are Bernoulli and generalized Bernoullifunctions, respectively. We employ the Fourier series technique to demonstrate reciprocity formulas for this sum. Derivedformulas are analogues of Mikolás’ reciprocity formula. Moreover, we offer Petersson–Knopp type identities for this sum.