Abstract
Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.
Highlights
The sawtooth function, denoted by ((x)), is defined by ⎧ (x) = ⎨x – ⎩0, [x], if x ∈/ Z, if x ∈ Z, (1)where [x] denotes the greatest integer function not exceeding x
As a further generalization of Apostol’s Dedekind sums, we study poly-Dedekind sums associated with poly-Bernoulli functions of index k, which are given by
It was shown by Dedekind that they satisfy the following reciprocity relation: 1h 1 m 1
Summary
The type 2 poly-Bernoulli polynomials of index k are defined by When x = 0, B(nk) = B(nk)(0), (n ≥ 0) are called the type 2 poly-Bernoulli numbers of index k. In [6, 7, 12], the poly-Bernoulli polynomials of index k are defined by the generating function When x = 0, βn(k) = βn(k)(0) are called the poly-Bernoulli numbers of index k.
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