Turkish Journal of Mathematics
DOI
10.3906/mat-1911-93
Abstract
In this paper, we are interested in higher-order character Dedekind sum% \[ \sum\limits_{v=0}^{ck-1}\chi_{1}\left( v\right) \mathcal{B}_{p,\chi_{2}% }\left( a\frac{v+z}{c}+x\right) \mathcal{B}_{q}\left( b\frac{v+z}% {ck}+y\right) ,\text{ }a,b,c\in\mathbb{N} \text{ and }x,y,z\in\mathbb{R}, \] where $\chi_{1}$ and $\chi_{2}$ are primitive characters of modulus $k,$ $\mathcal{B}_{p}\left( x\right) $ and $\mathcal{B}_{p,\chi_{2}}\left( x\right) $ 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 Mikolas' reciprocity formula. Moreover, we offer Petersson--Knopp type identities for this sum.
Keywords
Dedekind sum, Bernoulli polynomials, Fourier series
First Page
998
Last Page
1015
Recommended Citation
CAN, MÜMÜN
(2020)
"Higher-order character Dedekind sum,"
Turkish Journal of Mathematics: Vol. 44:
No.
3, Article 27.
https://doi.org/10.3906/mat-1911-93
Available at:
https://journals.tubitak.gov.tr/math/vol44/iss3/27
