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Some identities and reciprocity relationsof unipoly-Dedekind type DC sums
Journal of Inequalities and Applications volume 2021, Article number: 122 (2021)
Abstract
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.
1 Introduction
Let
where \([ \cdot ]\) denotes the greatest integer not exceeding x.
The Dedekind sums are defined by
where h and m are positive integers.
The most fundamental result in the theory of Dedekind sums is the reciprocity theorem. If h and m are relatively prime positive integers, then
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.
The Euler polynomials \(E_{n}(x)\) (\(n\in \mathbb{N}\cup \{0\}\)) are defined by their generating functions as follows:
When \(x=0\), \(E_{n}:=E_{n}(0)\) are called Euler numbers. The first few of Euler numbers are \(E_{0}=1\), \(E_{1}=-\frac{1}{2}\), \(E_{2}=0\), \(E_{3}=\frac{1}{4}\), \(E_{4}=0, \ldots \) , and \(E_{2n}=0\), (\(n\in \mathbb{N} \)).
The Genocchi polynomials \(G_{n}(x)\), (\(n\in \mathbb{N}\cup \{0\}\)), are defined by their generating functions as follows:
When \(x=0\), \(G_{n}: =G_{n}(0)\) are called Genocchi numbers. The first few of Euler numbers are \(G_{0}=0\), \(G_{1}=1\), \(G_{2}=-1\), \(G_{3}=0\), \(G_{4}=1, \ldots \) , and \(G_{2n+1} = 0\), (\(n\in \mathbb{N}\)).
We note that
From (2), we note that
and
with the usual convention about replacing \(E^{n}\) and \(G^{n}\) with \(E_{n}\) and \(G_{n}\), respectively.
From (2), for \(n\equiv 1 (\mathrm{{mod} 2)}\), we have
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 \(n\geq 0\).
The Euler function is defined by
where \([x]\) denotes the greatest integer not exceeding x.
Kim and Kim considered the modified polyexponential function defined by
Note that \(\mathrm{Ei}_{1}(x)=e^{x}-1\).
We introduce the type 2 poly-Euler polynomials, which are given by
When \(x=0\), \(E_{k,n}=E_{k,n}(0)\), \(n\ge 0\), are called type 2 poly-Euler numbers.
We also introduce the type 2 poly-Genocchi polynomials, which are given by
When \(x=0\), \(G_{k,n}=G_{k,n}(0)\), \(n\ge 0\), are called type 2 poly-Genocchi numbers.
By (12), we easily get \(G_{k,0}=0\), \(G_{k,1}=1\), \(G_{k,2}=-2+2^{l-k},\ldots\) . Since \(\mathrm{Ei}_{1}(\log (1+2t)) = 2t\), we see that \(E_{1,n}(x) = E_{n}(x)\) and \(G_{1,n}(x) = G_{n}(x)\) (\(n \geq 0\)) are the Euler polynomials and the Genocchi polynomials, respectively.
Kim introduced the Dedekind type DC sums given by
where \(\overline{E_{p}}(x)\) is the pth Euler function.
For \(p\in \mathbb{N}\) with \(p\equiv 1\) (mod2), the reciprocity law of \(T_{p}(h,m)\) is given by
where h, k are relative prime positive integers and
Recently, Ma et al. introduced the poly-Dedekind type DC sums associated with the type 2 poly-Euler functions, which are given by
where \(\overline{E}_{p}(h\mu /m)=E_{p} (\langle h\mu /m\rangle )\).
For \(n\in \mathbb{N}\cup \{0\}\), as is well known, the Stirling numbers of the first kind are defined by
and
For \(n\geq 0\), the Stirling numbers of the second kind are defined by
and
where \((x)_{0}=1\), \((x)_{n}=x(x-1)\dots (x-n+1)\), (\(n\geq 1\)).
2 Type 2 unipoly-Euler numbers and type 2 unipoly-Genocchi numbers
Let Ï„ be any arithmetic function which is real or complex valued and defined on the set of positive integers \(\mathbb{N} \). Then Kim and Kim defined the unipoly function attached to polynomials Ï„ by
When \(\tau (n)=1\), \(u_{k}(x|1)= \sum_{n=1}^{\infty } \frac{x^{n}}{n^{k}} =Li_{k}(x)\) is the ordinary polylogarithm function.
From (17), we have
Lee et al. introduced the type 2 unipoly-Euler polynomials of index k defined by
When \(x=0\), \(\mathcal{E}_{k,n, \tau }:= \mathcal{E}_{k,n, \tau }(0)\) are called type 2 unipoly-Euler numbers.
The type 2 unipoly-Genocchi polynomials of index k are defined by
When \(x=0\), \(\mathbb{G}_{k,n, \tau }:=\mathbb{ G}_{k,n, \tau }(0)\) are called type 2 unipoly-Genocchi numbers.
For \(n \in \mathbb{N}\cup \{0\}\) and \(k\in \mathbb{Z}\), let \(\tau (n)=\frac{1}{\Gamma (n)}=\frac{1}{(n-1)!}\). Then, from (12) and (19), we have
Thus, from (21) we have
Similarly, we get
Moreover, from (21), we note that
Thus, by (23), we get
In the same way as (23), we have
Furthermore, by (24) and (25), we have
Theorem 1
For \(n\ge 1\), we have
and
Proof
From (2), (15), and (19), we have
Therefore, from (29) we get identity (27).
By using (3), (15), and (19) in the same way as (29), we also get identity (28). □
In particular, we get
and the first few of the type 2 unipoly-Euler numbers are \(\mathcal{E}^{(k)}_{0,\tau }=\tau (1)\), \(\mathcal{E}^{(k)}_{1,\tau }= \frac{1}{2}\tau (1)+\frac{1}{2^{k-1}}\tau (2), \ldots \) . In addition,
and the first few of the type 2 unipoly-Genocchi numbers are \(\mathbb{G}_{k,0,\tau }=0\), \(\mathbb{G}_{k,1,\tau }= \tau (1), \ldots \) .
Since \(\mathbb{G}_{k,0,\tau }=0\), we note that
Thus, from (30), we have
Lemma 2
For \(n\ge 1\), 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). □
Theorem 3
For an odd positive integer \(d\geq 3\) and \(n\geq 1\), we have
and
Proof
For an odd positive integer \(d\geq 3\), from (15) and (17), we have
On the other hand,
Therefore, by (38) and (39), we obtain identity (36).
Moreover, from (31), we get identity (37). □
Theorem 4
For an odd positive integer \(d\geq 1\) and \(n\geq 0\), we have
Proof
From \(\sum_{i=0}^{n-1}(-1)^{i} e^{it} = \frac{(-1)^{n-1}e^{nt} +1}{e^{t} +1}\), \((n\equiv 1 (\mathrm{{mod} 2))}\), (2), and (13), we have
Therefore, by (40), we get the desired result. □
Corollary 5
For an odd positive integer \(d\geq 1\) and \(n\geq 1\), we have
Proof
From (31) and Theorem 4, we have
There, from (41), we arrive at the desired result. □
Lemma 6
For \(l\in \mathbb{N}\) with \(l< q\), we have
Proof
For \(l\in \mathbb{N}\) with \(l< q\), we have
On the other hand, by (24), we get
Therefore, by (42) and (43), we obtain what we want. □
Lemma 7
For \(q\in \mathbb{N}\), we have
Proof
By using (26), we observe that
On the other hand, by using (24), we have
3 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 \(h,m,q\in \mathbb{N}\) with \(q\equiv 1\) \((\mathrm{mod} 2)\) and \(\overline{\mathcal{E}}_{k,q,\tau }(x)=\mathcal{E}_{k,q,\tau }(x-[x])\) are the type 2 unipoly-Euler functions of index k (\([x]\) is the largest integer less than x).
For \(n \in \mathbb{N}\cup \{0\}\) and \(k\in \mathbb{Z}\), when \(\tau (n)=\frac{1}{\Gamma (n)}=\frac{1}{(n-1)!}\), we note that \(Z_{k,q,\frac{1}{\Gamma }}(h,m) = T_{q}^{(k)}(h,m)\).
In addition, we note that
Theorem 8
Let m be an odd positive integer ≥1 and \(q\in \mathbb{N}\). 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 \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Proof
For an odd integer \(q\geq 3\), we observe that \(E_{q-1}=0\). Moreover, \(E_{0}=1\) and \(\mathcal{E}_{k,0,\tau }=\tau (1)\). 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 \(E_{n}(x)=(E+x)^{n}\), \(\mathcal{E}_{k,n,\tau }(x)= (\mathcal{E}_{k,\tau }+x )^{n}\), (\(n\ge 0\)), with the usual convention about replacing \(E^{n}\) and \((\mathcal{E}_{k,\tau })^{n} \) with \(E_{n}\) and \(\mathcal{E}_{k,n,\tau }\), respectively.
Theorem 10
Let h, m be relatively prime positive integers and m, q be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Proof
As the index α ranges through the values \(\alpha =0,1,2,\ldots,m-1\), the product hα ranges over a complete residue system modulo m such that h, m are relatively prime positive integers, and we may replace \(<\frac{h\alpha }{m}>=\frac{h\alpha }{m}-[\frac{h\alpha }{m}]\) with \(<\frac{h\alpha }{m}>\) 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 \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Proof
From Corollary 5, we note that
where \(d\in \mathbb{N}\) with \(d\equiv 1\) (mod2), \(k\in \mathbb{Z}\), and \(n\ge 0\).
Therefore, from (52), we obtain the reciprocity relation for the type 2 unipoly-Dedekind type DC sums. □
Corollary 12
Let m, q, be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we get
where \(\Gamma (n) = (n-1)! \).
4 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
Note that \(Li_{1}(x)=-\log (1-x)\).
We consider the poly-Euler polynomials given by
When \(x=0\), \(E_{k,n}^{*}=E_{k,n}^{*}(0)\), \(n\ge 0\), are called poly-Euler numbers.
We also introduce the poly-Genocchi polynomials, which are given by
When \(x=0\), \(G_{k,n}^{*}=G_{k,n}^{*}(0)\), \(n\ge 0\), are called poly-Genocchi numbers.
Since \(Li_{1}(1-e^{-2t}) = 2t\), we see that \(E_{n}^{*(1)}(x) = E_{n}(x)\) and \(G_{n}^{*(1)}(x) = G_{n}(x)\) (\(n \geq 0\)) are the Euler polynomials and the Genocchi polynomials, respectively.
In addition, we define the unipoly-Euler polynomials of arbitrary index k defined by
When \(x=0\), \(E^{*}_{k,n, \tau }= E^{*}_{k,n, \tau }(0)\) are called unipoly-Euler numbers.
When \(\tau (n)=1\) for all n, \(E_{k,n,1}^{*}(x) = E_{k,n}^{*}(x)\) is the poly-Euler polynomials.
The unipoly-Genocchi polynomials of arbitrary index k are defined by
When \(x=0\), \(\mathbb{G}^{*}_{k,n, \tau }=\mathbb{G}^{*}_{k,n, \tau }(0)\) are called unipoly-Genocchi numbers.
When \(\tau (n)=1\) 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 \(h,m,q\in \mathbb{N}\) with \(q\equiv 1\) (mod2) and \(\overline{E}_{k,q,\tau }^{*}(x)=E_{k,q,\tau }^{*}(x-[x])\) are the unipoly-Euler functions of index k (\([x]\) is the largest integer less than x).
For \(n \in \mathbb{N}\cup \{0\}\) and \(k\in \mathbb{Z}\), let \(\tau (n)=1\). From \(\overline{E}_{k,q,1}^{*}=\overline{E}_{k,q}^{*}\) and (13), we note that \(Y_{k,q,1}(h,m) = T_{k,q}(h,m)\).
We note that
Thus, by (58), we get
In the same way as (59), we have
Furthermore, by (59) and (60), we have
Theorem 13
For \(n\ge 1\), we have
and
Proof
From (2), (16), and (55), we have
Therefore, from (64), we get identity (62).
By using (3), (16), and (56), in the same way as (64), we also get identity (63). □
In (62), when \(x=1\), we get
and the first few of the unipoly-Euler numbers are \(E^{*}_{k,0,\tau }=\tau (1)\), \(E^{*}_{k,1,\tau }= -\tau (1)+ \frac{1}{2^{k-1}}\tau (2), \ldots \) .
In (63), when \(x=1\), we get
and the first few of the unipoly-Genocchi numbers are \(G^{*}_{k,0,\tau }=0\), \(G^{*}_{k,1,\tau }= \tau (1), \ldots \) .
Since \(G_{k,0,\tau }^{*}=0\), we note that
Lemma 14
For \(n\ge 1\), we have
and
Proof
On the other hand, from (17), we have
Therefore, by (70) and (71), we obtain identity (68). By using (67), we get identity (69). □
Theorem 15
For an odd positive integer \(d\geq 3\) and \(n\geq 1\), we have
and
Proof
For an odd positive integer \(d\geq 3\), from (15) and (17), we have
On the other hand, from \(\sum_{i=0}^{n-1}(-1)^{i} e^{it} = \frac{(-1)^{n-1}e^{nt} +1}{e^{t} +1}\), \((n\equiv 1 (\mathrm{{mod} 2))}\), we have
Therefore, by (74) and (75), we obtain identity (72).
Moreover, from (67), we get identity (73). □
Theorem 16
For an odd positive integer \(d\geq 1\) and \(n\geq 0\), we have
Proof
From (2) and (14), for an odd positive integer \(d\geq 1\), we have
Therefore, by (76), we get the desired result. □
Corollary 17
For an odd positive integer \(d\geq 1\) and \(n\geq 1\), we have
Proof
From (67) and Theorem 16, we have
There, from (77), we arrive at the desired result. □
We can obtain the following lemmas in the same way as Lemma 6 and Lemma 7, respectively, in Sect. 2.
Lemma 18
For \(l\in \mathbb{N}\) with \(l< q\), we have
Lemma 19
For \(q\in \mathbb{N}\), we have
In addition, we can obtain the following theorems in the same way as Theorem 9 and Theorem 10, respectively, in Sect. 3.
Theorem 20
Let m, q be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Theorem 21
Let h, m be relatively prime positive integers and m, q be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Now, we obtain the following reciprocity theorem for the unipoly-Dedekind type DC sums associated with the unipoly-Euler function with index k.
Theorem 22
Let m, q be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we have
Proof
From Corollary 17, we note that
where \(d\in \mathbb{N}\) with \(d\equiv 1\) (mod2), \(k\in \mathbb{Z}\), and \(n\ge 0\).
From (78), in the same way as (53),
Therefore, from (79), we obtain the reciprocity relation for the unipoly-Dedekind type DC sums. □
Corollary 23
Let m, q, be odd positive integers \(m\geq 1\) and \(q\geq 3\), respectively. Then we get
where \(\tau (n)=1\) for all n.
5 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.
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Acknowledgements
The authors would like to thank the referees for the detailed and valuable comments that helped improve the original manuscript in its present form. Also, the authors thank Jangjeon Institute for Mathematical Science for the support of this research.
Funding
This work was supported by the Basic Science Research Program, the National Research Foundation of Korea (NRF-2021R1F1A1050151).
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Kim, H.K., Lee, D.S. Some identities and reciprocity relationsof unipoly-Dedekind type DC sums. J Inequal Appl 2021, 122 (2021). https://doi.org/10.1186/s13660-021-02655-2
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DOI: https://doi.org/10.1186/s13660-021-02655-2