Identities on poly-Dedekind sums

Dedekind sums occur in the transformation behaviour 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 poly-Dedekind sums which are 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.


Introduction
In order to give concise definition of the Dedekind sums, we introduce the notation (1) ( where [x] denotes the greatest integer not exceeding x. It is well known that the Dedekind sums are defined by , (see [6,7]).
In [5], the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k as (9) Ei k (log(1 + t)) e t − 1 .
n (0), (n ≥ 0), are called the type 2 poly-Bernoulli numbers of index k. Note here that B (1) n (x) = B n (x) are the Bernoulli polynomials. The fractional part of x is denoted by (10) x The Bernoulli functions are defined by [1,3,12]).
Thus, by (3) and (11), we get where h, m are relatively prime positive integers. We need the following lemma which is well-known or easily shown.
Lemma 1. Let n be a nonnegative integer, and let d be a positive integer. Then we have Dedekind showed that the quantity S(h, m) = ∑ m−1 µ=1 µ m B 1 hµ m occurs in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, he showed the following reciprocity relation for Dedekind sums: if h and m are relatively prime positive integers. Apostol considered the generalized Dedekind sums which are given by and showed in [1] that they satisfy the reciprocity relation In this paper, we consider the poly-Dedekind sums which are defined by p (x) are the type 2 poly-Bernoulli polynomials of index k (see (38)) and B (k) p ( x ) are the type 2 poly-Bernoulli functions of index k. Note here that S (1) p (h, m) = S p (h, m). We show the following reciprocity relation for the poly-Dedekind sums given by (see Theorem 10) For k = 1, this reciprocity relation for the poly-Dedekind sums reduces to that for the generalized Dedekind sums given by (see Corollary 11) In Section 2, we will derive various facts about the type 2 poly-Bernoulli polynomials that will be needed in the next section. In Section 3, we will define the poly-Dedekind sums and demonstrate a reciprocity relation for them.

On type 2 poly-Bernoulli polynomials
From (38), we note that Thus, by (14), we get By (15), we get From (38), we have On the other hand, where S 1 (n, m) are the Stirling numbers of the first kind. Therefore, by (17) and (18), we obtain the following theorem.
By Theorem 2, we get where δ n,k is the Kronecker's symbol.
With (16) in mind, we now compute On the other hand, by (15), we get Therefore, by (19) and (20), we obtain the following theorem.
Therefore, by Theorem 3 and (21), we obtain the following corollary.
On the other hand, by (15), we get Therefore, by (22) and (23), we obtain the following theorem.

Poly-Dedekind Sums
Apostol considered the generalized Dedekind sums which are given by In this section, we consider the poly-Dedekind sums which are given by p ( x ) are the type 2 poly-Bernoulli functions of index k. Note that Assume now that h = 1. Then we have From (5), we have By (26) and (27), we get Now, we assume that p is an odd positive integer ≥ 3, so that B p = 0. Then we have Therefore, by (29), we obtain the following proposition.
Proposition 6. Let p be an odd positive integer ≥ 3. Then we have We still assume that p is an odd positive integer ≥ 3, so that B p = 0. Then, from Corollary 4, Theorem 5 and Proposition 6, we note that To proceed further, we note that p i−2 p+1 i = 1 p+2 p+2 i (i − 1), for i ≥ 1, and that B (k) from Theorem 2. Then, from (30), we see that Therefore, by (31), we obtain the following theorem.
Theorem 7. For m ∈ N, and any odd positive integer p ≥ 3 , we have Now we employ the symbolic notation as Assume that h, m are relatively prime positive integers. Then we see that Now, as the index µ ranges over the values µ = 0, 1, 2, . . . , m − 1, the product hµ ranges over a complete residue system modulo m and due to the periodicity of B 1 (x), the term B 1 (hµ, m) may be replaced by B 1 (µ/m), without altering the sum over µ. Thus the sum (32) is equal to where we used the fact (a) in Lemma 1. Therefore, we obtain the following theorem.
Theorem 8. For m, n, h ∈ N, with (h, m) = 1, and p is any positive odd integer ≥ 3, we have where d is a positive integer. Therefore, by comparing the coefficients on both sides of (34), we obtain the following theorem.
Theorem 9. For k ∈ Z, d ∈ N and n ≥ 0, we have From (25), and by using Theorem 9 and (c) in Lemma 1, we see that Therefore, we obtain the following reciprocity relation.
Theorem 10. For m, h, p ∈ N and k ∈ Z, we have In case of k = 1, we obtain the following reciprocity relation for the generalized Dedekind sum defined by Apostol. In 1952, Apostol considered the generalized Dedekind sums and introduced interested and important identities and theorems related to his generalized Dedekind sums. These Dedekind sums are a field that has been studied by various researchers. Recently, the modified Hardy's polyexponential function of index k is introduced by x n n k (n − 1)! , (k ∈ Z), (see [5,10]).
In [5], the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k as In this paper, we thought of the poly-Dedekind sums from the perspective of the Apostol's generalized Dedekind sums. That is, we considered the poly=Dedekind sums which are derived from the type 2 poly-Bernoulli functions and polynomials.