poly-Dedekind type DC sums involving poly-Euler functions

The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

It is known that Dedekind type DC sums are given by , (see [14,20]).
is the polyexponential function of index k.
In [18], Komatsu defined the polylogarithm factorial function Li f k (x) by xLi f k (x) = xe(x, 1|k) = Ei k (x). So the polylogarithm factorial functions are special cases of Hardy's polyexponential functions, but our polyexponential functions are not. In fact, the slight difference between ours and Komatsu's is crutial in defining, for example, the type 2 poly-Bernoulli polynomials (see [10,15]) and also in constructing poly-Dedekind sums associated with such polynomials (see [16,19]). Here we recall from [10] that the type 2 poly-Bernoulli polynomials β (k) n (x) of index k are defined by We also recall that, for any integer k, the poly-Bernoulli polynomials B (k) n (x) of index k are defined by where the polylogarithm functions Li k (x) are given by Li k (x) = ∑ ∞ n=1 x n n k . The reason why Ei k (x) is needed and Li f k (x) is not in (12) is twofold. The first reason is that Ei k (x) has order 1, so that the composition Ei k (log(1 + t)) still has order 1, which is definitely required, whereas Li f k (x) has order 0, so that Li f k (log(1 + t)) also has order 0. The second reason is that we want β (1) n (x) to be the ordinary Bernoulli polynomials when k = 1. Indeed, Ei 1 (x) = e x − 1, so that β (1) n (x) are those polynomials with β (1) The construction of the type 2 poly-Bernoulli polynomials are in parallel with that of the poly-Bernoulli polynomials. We note Li k (x) has order 1, so that the composition Li k (1 − e −t ) also has order 1. In addition, Li 1 (x) = − log(1 − x), so that B (1) n (x) are the ordinary Bernoulli polynomials with B (1) (13)). Thus we may say that Ei k (x) is a kind of a compositional inverse type to Li k (x). Now, we define the poly-Euler polynomials of index k by n (0) are called the poly-Euler numbers of index k. Note that E (1) n (x) = E n (x) and G (1) n (x) = G n (x). From (11), we note that Apostol considered the generalized Dedekind sums which are given by and showed that they satisfy a reciprocity relation in [1,2].
) are the Bernoulli functions with B p (x) the Bernoulli polynomials given by We remark that the Dedekind sum S(h, m) = S 1 (h, m) appears in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group and a reciprocity law of that was demonstrated by Dedekind in 1892.
As an extension of the sums in (16), the poly-Dedekind sums, which are given by were considered and a reciprocity law for those sums was shown in [16,19]. Here B The Dedekind type DC sums (see (5)) were first introduced and shown to satisfy a reciprocity relation in [14]. The aim of this paper is to introduce the poly-Dedekind type DC sums (see (17)), which are obtained from the Dedekind type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation (see (18)). The motivation of this paper is to explore our new sums in connection with modular forms, zeta fuctions and trigonometric sums, just as in the cases of Apostol-Dedekind sums, their generalizations and of some related sums. Indeed, Simsek [20] found trigonometric representations of the Dedekind type DC sums and their relations to Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), and Hardy-Berndt sums. In addition, Bayad and Simsek [3] studied three new shifted sums of Apostol-Dedekind-Rademacher type. These sums generalize the classical Dedekind-Rademacher sums and can be expressed in terms of Jacobi modular forms or cotangent functions or special values of the Barnes multiple zeta functions. They found reciprocity laws for these sums and demonstrated that some well-known reciprocity laws can be deduced from their results. We plan to carry out this line of research in a subsequent paper.
In this paper, we consider the poly-Dedekind type DC sums defined by where h, m, p ∈ N, and E (k) p are the poly-Euler functions of index k given by E . We show the following reciprocity relation for the poly-Dedekind type DC sums given by where m, h, p ∈ N with m ≡ 1 (mod 2), h ≡ 1 (mod 2), and k ∈ Z.
For k = 1, this reciprocity relation for the poly-Dedekind type DC sums reduces to that for the Dedekind type DC sums given by (see Corollary 15) (5) and (17), we see that the poly-Dedekind type DC sums are obtained from the Dedekind type DC sums by replacing the Euler functions by poly-Euler functions of arbitrary indices. Here we have to observe that the key to this generalization is the construction of poly-Euler polynomials defined in (14), which is done in an elaborate manner. First, we replace t by Ei k log(1 + t) as in (9), so that we construct the poly-Genocchi polynomials G

Remark 2. From
n (x) = G n (x) are the usual Genocchi polynomials. Next, by defining the poly-Euler polynomials E (k) n (x) as in (14), so that we have the desirable property E (1) n (x) = E n (x). Consequently, for k = 1, the poly-Dedekind type DC sums T In Section 2, we will derive various facts about the poly-Genocchi polynomials and poly-Euler polynomials that will be needed in the next section. In Section 3, we will define the poly-Dedekind type DC sums and demonstrate, among other things, a reciprocity relation for them.

POLY-GENOCCHI POLYNOMIALS AND POLY-EULER POLYNOMIALS
By (9), we have On the other hand, we also have where S 1 (n, m) are the Stirling numbers of the first kind. Therefore, by (19) and (20), we get the following theorem.   From (11) and (15), we see that Thus, we note that From (2) and (8), we have On the other hand, we also have Therefore, by (21) and (22), we obtain the following theorem.
On the other hand, we also have Therefore, by (25) and (26), we obtain the following theorem.
Theorem 6. For x, n ∈ N, we have Note that, for k = 1, we have Corollary 7. For x, n ∈ N, we have Note that For m ∈ N with m ≡ 1 (mod 2), we note that Therefore, by (27), we obtain the following theorem.
Theorem 8. For n ≥ 0, and m ∈ N with m ≡ 1 (mod 2), we have From Theorem 6, we have where n, m ∈ N with m ≡ 1 (mod 2). Thus we obtain the important corollary that will be used in deriving the reciprocity law in Theorem 14.
Corollary 9. For n, m ∈ N with m ≡ 1 (mod 2), we have Note that where n, m ∈ N with m ≡ 1 (mod 2). For p, s ∈ N with s < p, we have On the other hand, by (15), we get Therefore, by (28) and (29), we obtain the following lemma.
Lemma 10. For p, s ∈ N with s < p, we have In particular, for k = 1, and p, s ∈ N with p ≡ 1 (mod 2), s ≡ 0 (mod 2), we have We note that On the other hand, from (15), we have Therefore, by (30) and (31), we obtain the following lemma.
Lemma 11. For p ∈ N, we have In particular, for p ∈ N with p ≡ 1 (mod 2), and k = 1, we get

POLY-DEDEKIND TYPE DC SUMS
The Dedekind type DC sums are defined by where E p (x) is the p-th Euler function (see [14,20]). For p ∈ N with p ≡ 1 (mod 2), and relative prime positive integers m, h with m ≡ 1 (mod 2), h ≡ 1 (mod 2), the reciprocity law of T p (h, m) is given by where µ runs over all integers satisfying 0 ≤ µ ≤ m − 1 and µ − hµ m ≡ 1 (mod 2), and For the rest of our discussion, we assume that k is any integer. In light of (32), we define poly-Dedekind type DC sums given by where h, m, p ∈ N, and E (k) p are the poly-Euler functions given by By (32) and (33), we get Let us take h = 1. Then we have Now, we assume that p ≥ 3 is an odd integer, so that E p−1 = 0. Interchanging the order of summation in (39), we have Therefore, we obtain the following theorem.
Theorem 13. For m ∈ N with m ≡ 1 (mod 2), and p ≡ 1 (mod 2) with p > 1, we have In other words, we have We observe that From (42) and Lemma 8, we have By (43), we get From (41) and (44), we note that It is easy to show that (45) and (46), we get Therefore, by (47), we obtain the following theorem.
Theorem 14. For m ∈ N with m ≡ 1 (mod 2), p ≡ 1 (mod 2) with p > 1, we have Now, we employ the symbolic notations as E n (x) = (E + x) n , E were considered and shown to satisfy a reciprocity relation in [16], where B  were introduced and shown to satisfy a reciprocity relation in [14], where E p (x) is the p-th Euler function.
Simsek found trigonometric representations of the Dedekind type DC sums and their relations to Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), and Hardy--Berndt sums. In this paper, as a furthrer generalization of the Dedekind type DC sums, the poly-Dedekind type DC sums given by