Poly-Dedekind sums associated with poly-Bernoulli functions

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.

The modified Hardy's polyexponential function of index k is defined by x n n k (n -1)! (k ∈ Z), (see [7]).
In [6,7,12], the poly-Bernoulli polynomials of index k are defined by the generating function When x = 0, β (k) n = β (k) n (0) are called the poly-Bernoulli numbers of index k.
where h, m are relatively prime positive integers. Apostol considered the generalized Dedekind sums, which are given by and showed in [1,2] that they satisfy the reciprocity relation As one generalization of Apostol's generalized Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k were recently introduced (see [13]) and, among other things, a reciprocity relation for them was derived.
In this paper, as another generalization of Apostol's generalized Dedekind sums, we consider the poly-Dedekind sums defined by are the poly-Bernoulli functions of index k (see (10)). Note here that T (1) p (h, m) = S p (h, m). We show the following reciprocity relation for the poly-Dedekind sums given by (see Theorem 7) For k = 1, this reciprocity relation for the poly-Dedekind sums reduces to that for Apostol's generalized Dedekind sums given by (see Corollary 8) We recommend the readers to look at the articles [15,16,18,21] and the more recent one [14], which are related to the present paper. In Sect. 2, we derive various facts about the poly-Bernoulli polynomials that will be needed in the next section. In Sect. 3, we define the poly-Dedekind sums associated with the poly-Bernoulli functions and demonstrate a reciprocity relation for them.

Poly-Dedekind sums associated with poly-Bernoulli functions
Let n be a nonnegative integer. Then the Stirling numbers of the second kind are defined by [1-14, 17, 19] From (9) and (10), we note that Thus, by (15), we get On the other hand, Therefore, by (16) and (17), we obtain the following theorem.
Theorem 1 For n ∈ N, we have From Theorem 1, we note that Taking k = 1 in Theorem 1 gives us the following corollary. where δ n,k is the Kronecker symbol.
The three identities in the following lemma can be shown just as in Theorem 3, Corollary 4, and Theorem 5 of [13], and hence their proofs are left to the reader as exercises.
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 where h, m, p ∈ N, k ∈ Z, and β (k) The two identities in Lemma 4 can be proved in the same way as in Proposition 6 and Theorem 7 in [13], while the identity in Lemma 5 can be shown just as in Theorem 8 in [13]. Therefore their proofs are left to the reader.

Lemma 5
For m, n, h ∈ N with (h, m) = 1, and p any positive odd integer ≥ 3, we have For d ∈ N, we observe that Therefore, by (19), we obtain the following theorem.

Theorem 6
For k ∈ Z, d ∈ N, and n ≥ 0, we have By (18), Lemmas 3-5, and Theorem 6, we get Therefore, by (20), we obtain the following reciprocity theorem for the poly-Dedekind sums associated with poly-Bernoulli functions with index k.
Theorem 7 For m, h, p ∈ N and k ∈ Z, we have In case of k = 1, by making use of Corollary 2, we obtain the following reciprocity relation for the generalized Dedekind sums defined by Apostol.

Conclusion
The quantity called the Dedekind sum, occurs in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. It was shown by Dedekind that they satisfy the following reciprocity relation: