Multiple Dedekind Type Sums and Their Related Zeta Functions

Abdelmejid Bayad,Abdelmejid Bayad,Abdelmejid Bayad,Abdelmejid Bayad,Yilmaz Simsek

doi:10.3390/math9151744

Abstract

The main purpose of this paper is to use the multiple twisted Bernoulli polynomials and their interpolation functions to construct multiple twisted Dedekind type sums. We investigate some properties of these sums. By use of the properties of multiple twisted zeta functions and the Bernoulli functions involving the Bernoulli polynomials, we derive reciprocity laws of these sums. Further developments and observations on these new Dedekind type sums are given.

Highlights

; Hn(u) are rational fractions of polynomials and were studied in great detail by Frobenius [13], who was interested in their relationship to Bernoulli numbers [14]
The above Theorem 1 and the results of Chapter II of Koblitz’s book [8] illustrate that the twisted (h, q)-Bernoulli polynomials are p-adic in essence and have profound connections with the special values of certain zeta functions
Let us specify the definition of the twisted (h, q)-Bernoulli numbers of order v by using the following generating function:

Summary

One can observe that for ξ

The twisted (h, q)-Bernoulli polynomials are defined in [9] by using the generating function. The twisted (h, q)-Bernoulli numbers and polynomials of order v are given in [1,10,12]) by their generating functions:. The above Theorem 1 and the results of Chapter II of Koblitz’s book [8] illustrate that the twisted (h, q)-Bernoulli polynomials are p-adic in essence and have profound connections with the special values of certain zeta functions. Let us specify the definition of the twisted (h, q)-Bernoulli numbers of order v by using the following generating function:. Log qha + at vexyt ((ξqh)aeat − 1)v. By using (4), we have b1,··· ,by−1≥0 b1 +···+by−1 =v v b1, · · · , by−1 av ξaqha

By identifying the coefficients of tn n!

We easily see from the above that

We note that it to prove

Let n