P-Adic q-Twisted Dedekind-Type Sums

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

doi:10.3390/sym13091756

Abstract

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters.

Highlights

For a positive integer h and an integer k, the classical Dedekind sum is defined as k−1 s(h, k) = ∑ a=1 a k ha k, Academic Editor: José Carlos R
The main result of this paper is to prove their Dedekind reciprocity laws
Abn+2S(ph,ξ)(s + 2; a, b : q) + ban+2S(ph,ξ)(s + 2; b, a; q) is a continuous function of s on Zp and for all positive integer n such that n + 1 ≡ 0 (mod (p − 1)) this reciprocity law reduces to the reciprocity law for higher-order q-Dedekind type sum which is given by authors [18]: abnSn(h,ξ)(a, b : q) + banSn(h,ξ)(b, a : q)

Summary

Introduction

Received: 16 June 2021 Accepted: 26 July 2021 Published: 20 September 2021 where ((x)) =. Dedekind [1] introduced this sum in connection with the transformation formula for, the well-known modular form of weight. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. From this transformation formula, for h and k are coprime integers, Dedekind deduced his reciprocity formula s(h, k) + s(k, h) = − 1 + 1 4 12 h+ 1 +k k kh h (1). For odd values of m, these higher-order Dedekind sums enjoy a reciprocity law, first proved by Apostol [3]:. In 1953, Carlitz [4] generalized sm(h, k) by defining the higher order Dedekind sums as

Br a k

Findings

Since s k