Identities on poly-Dedekind sums

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.

Recently, Kim and Kim [5,9] considered the polyexponential function of index k given by x n n k (n -1)! (k ∈ Z).
In [5] the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k as When x = 0, B (k) n = B (k) n (0) (n ≥ 0) are called the type 2 poly-Bernoulli numbers of index k. Note that B (1) n (x) = B n (x) are the Bernoulli polynomials. The fractional part of x is denoted by The Bernoulli functions are defined by [1,4,11]).
Thus by (3) and (11) we get where h, m are relatively prime positive integers.
We need the following lemma, which is well-known and 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 behavior 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 [1] considered the generalized Dedekind sums given by and showed that they satisfy the reciprocity relation In this paper, we consider the poly-Dedekind sums defined by where B (k) p (x) are the type 2 poly-Bernoulli polynomials of index k (see (9)), and B (k) . We show the following reciprocity relation for the poly-Dedekind sums (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 Sect. 2, we derive various facts about the type 2 poly-Bernoulli polynomials, which will be needed in the next section. In Sect. 3, we define the poly-Dedekind sums and demonstrate a reciprocity relation for them.

On type poly-Bernoulli polynomials
Note that by (9) Thus by (14) we get By (15) we get From (9) 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.
Theorem 2 For n ≥ 1, we have By Theorem 2 we get where δ n,k is the Kronecker 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.
Corollary 4 For s, p ∈ N, we have 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 given by where B p (hμ/m) = B p ( hμ/m ). Note that, for any relatively prime positive integers h, m, we have In this section, we consider the poly-Dedekind sums given by where h, m, p ∈ N, k ∈ Z, and B Assume now that h = 1. Then we have

By (26) and (27) we get
Now we assume that p ≥ 3 is an odd positive integer, so that B p = 0. Then we have Therefore by (29) we obtain the following proposition.

Proposition 6 Let p ≥ 3 be an odd positive integer. Then we have
We still assume that p ≥ 3 is an odd positive integer, 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) 1 (1) -B (k) 1 = 1 by 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 where we used the fact (a) in Lemma 1. Therefore we obtain the following theorem.
In 1952, Apostol considered the generalized Dedekind sums and introduced interesting and important identities and theorems related to his generalized Dedekind sums. These Dedekind sums are a field studied by various researchers. Recently, the modified Hardy polyexponential function of index k is introduced by x n n k (n -1)! , (k ∈ Z) (see [5,9]).
In [5] the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k by Ei k (log(1 + t)) e t -1 e xt = ∞ n=0 B (k) n (x) t n n! (k ∈ Z).
In this paper, we thought of the poly-Dedekind sums from the perspective of the Apostol generalized Dedekind sums. That is, we considered the poly-Dedekind sums derived from the type 2 poly-Bernoulli functions and polynomials.