Higher-order character Dedekind sum

Abstract

In this paper, we are interested in higher-order character Dedekind sum ck∑−1 v=0 χ1 v Bp,χ2 a v + z c + x Bq b v + z ck + y , a, b, c ∈ N and x, y, z ∈ R, where χ1 and χ2 are primitive characters of modulus k, Bp x and Bp,χ2 x are Bernoulli and generalized Bernoulli functions, respectively. We employ the Fourier series technique to demonstrate reciprocity formulas for this sum. Derived formulas are analogues of Mikolás’ reciprocity formula. Moreover, we offer Petersson–Knopp type identities for this sum.