Reciprocity of poly-Dedekind-type DC sums involving poly-Euler functions

Abstract

The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind-type DC sums, which are obtained from the Dedekind-type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

“…Bayad and Simsek [4,28] studied three new shifted sums of Apostol-Dedekind-Rademacher type. The Dedekind type DC (Daehee and Changhee) sums using Euler functions were first introduced by Kim [13] and have been studied variously by several authors since then [20,30]. Recently, as a generalization of Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k [18] and the unipoly-Dedkind sum [11] were introduced.…”

“…Recently, as a generalization of Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k [18] and the unipoly-Dedkind sum [11] were introduced. In addition , Ma et al introduced the poly-Dedekind sums associated with the poly-Bernoulli functions of index k [21] and the poly-Dedekind type DC sums associated with the type 2 poly-Euler functions of index k [20].…”

“…The Euler polynomials E n (x) (n ∈ N ∪ {0}) are defined by their generating functions as follows: [13,14,19,20]).…”

“…, and E 2n = 0, (n ∈ N). The Genocchi polynomials G n (x), (n ∈ N∪{0}), are defined by their generating functions as follows: [9,20,29]).…”

“…with the usual convention about replacing E n and G n with E n and G n , respectively. From (2), for n ≡ 1(mod2), we have [20]).…”