A study on a type of degenerate poly-Dedekind sums

: Dedekind sums and their generalizations are de ﬁ ned in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.


Introduction and preliminaries
As is well known, the Euler polynomials and Bernoulli polynomials are given by ( ) ( ) ( )  x denotes the greatest integer function not exceeding x.
The poly-Dedekind-type DC sums, which are obtained by replacing the Euler functions appearing in Dedekind-type DC sums by any poly-Euler functions of arbitrary indices, were given by where

E x
p k are the poly-Euler functions, ( ) are the poly-Euler polynomials given by are the poly-Euler numbers.In this article, as another generalization, we consider degenerate poly-Dedekind sums, which are obtained by replacing the first Bernoulli function appearing in Dedekind sums by degenerate poly-Bernoulli functions of arbitrary indices.We derive a reciprocity relation for them.
For the rest of this section, we recall some necessary facts that are needed throughout this article.For any ∈ λ , the degenerate exponential function is defined by where In [19], the degenerate Bernoulli polynomials are defined by are the degenerate Bernoulli numbers.We observe that On the other hand, By ( 10) and ( 11), we have The type 2 degenerate Stirling numbers are defined by (see [20,21]) The degenerate polylogarithmic function of index k is defined by Kim and Kim to be (see [20]) .
where ( ) is the inverse to ( ) e x λ .In [16], the degenerate poly-Bernoulli polynomials of index k are defined by are the degenerate poly-Bernoulli numbers.From ( 16), we note that 2 Reciprocity relations for the degenerate poly-Dedekind sums By replacing poly-Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices from the poly-Dedekind sums, the degenerate poly-Dedekind sums are given in the following, where are the degenerate poly-Bernoulli functions, From ( 16), we note that

So we have
A study on a type of degenerate poly-Dedekind sums  3 On the other hand, we also have Therefore, by ( 21) and ( 22), we obtain the following theorem.
Note that, for = k 1, we have the following corollary.
Corollary 1.For ∈ n , we have where δ n k , is the Kronecker's symbol.
For ∈ d , we observe that Therefore, by (23), we obtain the following theorem.
Therefore, by (24), we obtain the following reciprocity theorem for the degenerate poly-Dedekind sums associated with degenerate poly-Bernoulli functions with index k.By ( 12) and ( 17), let = h 1, we note that numbers.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 functions, where [ ] Corollary 1, we obtain the following reciprocity relation for the generalized Dede- kind sums defined by Apostol.