Abstract
Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.
Highlights
To give concise definition of the Dedekind sums, we introduce the notation ((x)) = x [x] 1 2 if x ∈/ Z, (1)if x ∈ Z, where [x] denotes the greatest integer not exceeding x
In [5] the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k as
We show the following reciprocity relation for the poly-Dedekind sums: hmpSp(k)(h, m) + mhpSp(k)(m, h)
Summary
When x = 0, B(nk) = B(nk)(0) (n ≥ 0) are called the type 2 poly-Bernoulli numbers of index k. In 1892, he showed the following reciprocity relation for Dedekind sums: 1h 1 m 1 Apostol [1] considered the generalized Dedekind sums given by m–1 μ hμ We consider the poly-Dedekind sums defined by
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