Abstract
Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.
Highlights
Let ⎧ (x) = ⎨x – ⎩0, [x], if x ∈/ Z, if x ∈ Z,(see [1,2,3,4,5,6,7,8, 20,21,22,23,24,25,26,27,28, 30]), where [·] denotes the greatest integer not exceeding x
We introduce the type 2 poly-Euler polynomials, which are given by
Lee et al introduced the type 2 unipoly-Euler polynomials of index k defined by uk(log(1 + 2t)|τ t(et + 1)
Summary
2 Type 2 unipoly-Euler numbers and type 2 unipoly-Genocchi numbers Let τ be any arithmetic function which is real or complex valued and defined on the set of positive integers N. Lee et al introduced the type 2 unipoly-Euler polynomials of index k defined by uk(log(1 + 2t)|τ t(et + 1) The type 2 unipoly-Genocchi polynomials of index k are defined by uk(log(1 + 2t)|τ et + 1
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