Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

The most fundamental result in the theory of Dedekind sums is the reciprocity theorem. If h and m are relatively prime positive integers, then S(h, m) + S(m, h) = -1 4 + 1 12 hm , (see [2]).
It is well known that the classical Dedekind sums S(h, m), initiated by Richard Dedekind [9], first arose in the transformation formula of the logarithm of the Dedekind eta function. Dedekind introduced them to express the functional equation of the Dedekind eta function. These sums have figured prominently in so many different areas such as elliptic modular functions to number theory, analysis, number theory, combinatorics, q-series, Weierstrass elliptic functions, modular forms, and other areas [1-9, 13, 14, 16, 18, 20-30]. In combinatorial number theory, one is interested in partitions of an integer n from a finite set of positive integers. Beck et al. showed that the number of such partitions of n from a finite set is a quasi-polynomial in n, whose coefficients are built up from some generalization of Dedekind sums [3]. Bayad and Simsek [4,28] studied three new shifted sums of Apostol-Dedekind-Rademacher type. The Dedekind type DC (Daehee and Changhee) sums using Euler functions were first introduced by Kim [13] and have been studied variously by several authors since then [20,30]. Recently, as a generalization of Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k [18] and the unipoly-Dedkind sum [11] were introduced. In addition , Ma et al. introduced the poly-Dedekind sums associated with the poly-Bernoulli functions of index k [21] and the poly-Dedekind type DC sums associated with the type 2 poly-Euler functions of index k [20].
In this paper, we introduce two kinds of new generalizations of the poly Dedekind type DC sums. In Sect. 2, for our goal, we show explicit formulas of type 2 unipoly-Euler polynomials and type 2 unipoly-Genocchi polynomials. In Sect. 3, we introduce a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and derive the reciprocity relation for these unipoly-Dedekind type DC sums. In Sect. 4, we introduce a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and derive the reciprocity relation for those.
Let d be an odd positive integer ≥ 3. Then we have the following well-known relation: where d is an odd positive integer ≥ 3 and n ≥ 0. The Euler function is defined by , (n ≥ 0), (see [13,20,30] where [x] denotes the greatest integer not exceeding x. Kim and Kim considered the modified polyexponential function defined by x n n k (n -1)! , (k ∈ Z), (see [10]).
Kim introduced the Dedekind type DC sums given by where E p (x) is the pth Euler function.
For p ∈ N with p ≡ 1 (mod2), the reciprocity law of T p (h, m) is given by , (see [13,30] where h, k are relative prime positive integers and (Eh + Em) p = p l=0 p l E l h l E p-l m p-l (see [13,30]).

Type unipoly-Euler numbers and type unipoly-Genocchi numbers
Let τ be any arithmetic function which is real or complex valued and defined on the set of positive integers N. Then Kim and Kim defined the unipoly function attached to polynomials τ by , (see [10]). When Lee et al. introduced the type 2 unipoly-Euler polynomials of index k defined by u k (log(1 + 2t)|τ ) t(e t + 1) e xt = ∞ n=0 E k,n,τ (x) t n n! (see [19]).
In particular, we get and the first few of the type 2 unipoly-Euler numbers are and the first few of the type 2 unipoly-Genocchi numbers are G k,0, Thus, from (30), we have and Proof From (20), we have On the other hand, from (17), we have Therefore, by (34) and (35), we obtain identity (32). By using (31), we get identity (33).

Corollary 5
For an odd positive integer d ≥ 1 and n ≥ 1, we have Proof From (31) and Theorem 4, we have There, from (41), we arrive at the desired result.

Lemma 6
For l ∈ N with l < q, we have On the other hand, by (24), we get Therefore, by (42) and (43), we obtain what we want. .
Proof By using (26), we observe that On the other hand, by using (24), we have Therefore, by (44) and (45), we get what we want.

Unipoly-Dedekind type DC sums associated withthe type 2 unipoly-Euler functions of index k
In this section, as a generalization of the poly-Dedekind type DC sums, we consider the unipoly-Dedekind type DC sums associated with the type 2 unipoly-Euler functions of index k and derive several noble identities and the reciprocity relation for these. Naturally, we consider the unipoly-Dedekind type DC sums associated with the type 2 unipoly-Euler functions of index k as follows: where h, m, q ∈ N with q ≡ 1 (mod2) and E k,q,τ (x) = E k,q,τ (x -[x]) are the type 2 unipoly-Euler functions of index k ([x] is the largest integer less than x).
For n ∈ N ∪ {0} and k ∈ Z, when τ (n) = 1 (n) = 1 (n-1)! , we note that Z k,q, 1 (h, m) = T (k) q (h, m). In addition, we note that Theorem 8 Let m be an odd positive integer ≥ 1 and q ∈ N. Then we have Proof From (5), (7), and (46), we have By multiplying both sides of (47) by m q , we arrive at the desired result.

Theorem 9
Let m, q be odd positive integers m ≥ 1 and q ≥ 3, respectively. Then we have Proof For an odd integer q ≥ 3, we observe that E q-1 = 0. Moreover, E 0 = 1 and E k,0,τ = τ (1). From (24), we observe that In addition, by using Lemma 6, we have Therefore, from Theorem 8, (48), and (49), we get To prove the next theorem, we employ the symbolic notations as E n (x) = (E + x) n , E k,n,τ (x) = (E k,τ + x) n , (n ≥ 0), with the usual convention about replacing E n and (E k,τ ) n with E n and E k,n,τ , respectively.
Proof As the index α ranges through the values α = 0, 1, 2, . . . , m -1, the product hα ranges over a complete residue system modulo m such that h, m are relatively prime positive integers, and we may replace < hα m >= hα m -[ hα m ] with < hα m > without alternating the sum over α.
Therefore, from (8), we observe that Therefore, from (50), we obtain what we want.

Theorem 11
Let m, q be odd positive integers m ≥ 1 and q ≥ 3, respectively. Then we have Proof From Corollary 5, we note that where d ∈ N with d ≡ 1 (mod2), k ∈ Z, and n ≥ 0.

Unipoly-Dedekind type DC sums associated withunipoly-Euler functions of index k
In this section, as another generalization of the poly Dedekind type DC sums, we consider unipoly-Dedekind type DC sums associated with the unipoly-Euler functions of index k and derive the reciprocity relation for these. For the purposes of this section, we first introduce two new polynomials, the poly-Euler polynomials and poly-Genocchi polynomials, using the polylogarithm function of arbitrary index k.
It is well known that the polylogarithm function of index k is defined by x n n k , (k ∈ Z), (see [8,10,12]).
When τ (n) = 1 for all n, G * k,n,1 (x) = G * k,n (x) is the poly-Genocchi polynomials. Now, we consider a new type of unipoly-Dedekind type DC sums associated with the unipoly-Euler function of index k as follows: where h, m, q ∈ N with q ≡ 1 (mod2) and E * k,q,τ (x) = E * k,q,τ (x -[x]) are the unipoly-Euler functions of index k ([x] is the largest integer less than x).
We can obtain the following lemmas in the same way as Lemma 6 and Lemma 7, respectively, in Sect. 2.

Lemma 18
For l ∈ N with l < q, we have

Lemma 19
For q ∈ N, we have q j=0 q j In addition, we can obtain the following theorems in the same way as Theorem 9 and Theorem 10, respectively, in Sect. 3.

Corollary 23
Let m, q, be odd positive integers m ≥ 1 and q ≥ 3, respectively. Then we get m q Y 1,q,1 (h, m) + h q Y 1,q,1 (m, h) = m q T q (h, m) + h q T q (m, h) where τ (n) = 1 for all n.

Conclusion
In this paper, as further generalizations of the poly-Dedekind type DC sums, we introduced two kinds of unipoly-Dedekind type DC sums. In Sect. 3, the type 2 unipoly-Dedekind type DC sums associated with the type 2 unipoly-Euler functions of index k were introduced, and some interesting identities and the reciprocity relation were shown. In Sect. 4, the unipoly-Dedekind type DC sums associated with the unipoly-Euler functions of index k were introduced, and some interesting identities and the reciprocity relation were shown. We would like to further study another Dedekind type DC sums.