Abstract
Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.
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1 Introduction
The sawtooth function, denoted by \(( (x) )\), is defined by
where \([x]\) denotes the greatest integer function not exceeding x.
The Dedekind sums are defined by
where h is any integer and m is a positive integer (see [9–11, 17, 19, 20]).
It is well known that the Bernoulli polynomials are defined by
When \(x=0\), \(B_{n}=B_{n}(0)\), (\(n\ge 0\)) are called the Bernoulli numbers.
From (3), we note that
By (3), we easily get
and
The modified Hardy’s polyexponential function of index k is defined by
Note that \(\operatorname{Ei}_{1}(x)=e^{x}-1\).
Recently, the type 2 poly-Bernoulli polynomials of index k are defined by
When \(x=0\), \(B_{n}^{(k)}=B_{n}^{(k)}(0)\), (\(n\ge 0\)) are called the type 2 poly-Bernoulli numbers of index k. Note that \(B_{n}^{(1)}(x)=B_{n}(x)\), (\(n\ge 0\)).
It is well known that the polylogarithmic function of index k is defined by
Note that \(\operatorname{Li}_{1}(x)=-\log (1-x)\).
In [6, 7, 12], the poly-Bernoulli polynomials of index k are defined by the generating function
When \(x=0\), \(\beta _{n}^{(k)}= \beta _{n}^{(k)}(0)\) are called the poly-Bernoulli numbers of index k.
From (10), we note that
The fractional part of x is defined by
The Bernoulli functions are defined by
From (2), we have
where h, m are relatively prime positive integers.
Apostol considered the generalized Dedekind sums, which are given by
and showed in [1, 2] that they satisfy the reciprocity relation
As one generalization of Apostol’s generalized Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions of index k
were recently introduced (see [13]) and, among other things, a reciprocity relation for them was derived.
In this paper, as another generalization of Apostol’s generalized Dedekind sums, we consider the poly-Dedekind sums defined by
where \(\overline{\beta }_{p}^{(k)}(x)=\beta _{p}^{(k)}(\langle x \rangle )\) are the poly-Bernoulli functions of index k (see (10)). Note here that \(T_{p}^{(1)}(h,m)=S_{p}(h,m)\). We show the following reciprocity relation for the poly-Dedekind sums given by (see Theorem 7)
For \(k=1\), this reciprocity relation for the poly-Dedekind sums reduces to that for Apostol’s generalized Dedekind sums given by (see Corollary 8)
We recommend the readers to look at the articles [15, 16, 18, 21] and the more recent one [14], which are related to the present paper. In Sect. 2, we derive various facts about the poly-Bernoulli polynomials that will be needed in the next section. In Sect. 3, we define the poly-Dedekind sums associated with the poly-Bernoulli functions and demonstrate a reciprocity relation for them.
2 Poly-Dedekind sums associated with poly-Bernoulli functions
Let n be a nonnegative integer. Then the Stirling numbers of the second kind are defined by
where \((x)_{0}=1\), \((x)_{n}=x(x-1)\cdots (x-n+1)\), (\(n\ge 1\)).
From (9) and (10), we note that
Thus, by (15), we get
On the other hand,
Therefore, by (16) and (17), we obtain the following theorem.
Theorem 1
For \(n\in \mathbb{N}\), we have
From Theorem 1, we note that
Taking \(k=1\) in Theorem 1 gives us the following corollary.
Corollary 2
For \(n\in \mathbb{N}\), we have
where \(\delta _{n,k}\) is the Kronecker symbol.
The three identities in the following lemma can be shown just as in Theorem 3, Corollary 4, and Theorem 5 of [13], and hence their proofs are left to the reader as exercises.
Lemma 3
For \(s,p\in \mathbb{N}\), we have
and
As a further generalization of Apostol’s Dedekind sums, we study poly-Dedekind sums associated with poly-Bernoulli functions of index k, which are given by
where \(h,m,p\in \mathbb{N}\), \(k\in \mathbb{Z}\), and \(\overline{\beta }_{p}^{(k)}(x)=\beta _{p}^{(k)} (\langle x\rangle )\) are the poly-Bernoulli functions of index k.
Note that
The two identities in Lemma 4 can be proved in the same way as in Proposition 6 and Theorem 7 in [13], while the identity in Lemma 5 can be shown just as in Theorem 8 in [13]. Therefore their proofs are left to the reader.
Lemma 4
Let p be an odd positive integer ≥3, and \(m\in \mathbb{N}\). Then we have
and
Lemma 5
For \(m,n,h\in \mathbb{N}\) with \((h,m)=1\), and p any positive odd integer ≥3, we have
For \(d\in \mathbb{N}\), we observe that
Therefore, by (19), we obtain the following theorem.
Theorem 6
For \(k\in \mathbb{Z}\), \(d\in \mathbb{N}\), and \(n\ge 0\), we have
By (18), Lemmas 3–5, and Theorem 6, we get
Therefore, by (20), we obtain the following reciprocity theorem for the poly-Dedekind sums associated with poly-Bernoulli functions with index k.
Theorem 7
For \(m,h,p\in \mathbb{N}\) and \(k\in \mathbb{Z}\), we have
In case of \(k = 1\), by making use of Corollary 2, we obtain the following reciprocity relation for the generalized Dedekind sums defined by Apostol.
Corollary 8
For \(m,h,p \in \mathbb{N}\), we have
3 Conclusion
The quantity called the Dedekind sum,
occurs in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. It was shown by Dedekind that they satisfy the following reciprocity relation:
if h and m are relatively prime positive integers.
Apostol considered the generalized Dedekind sums
and derived a reciprocity relation for them. Recently, as one generalization of the generalized Dedekind sums, the poly-Dedekind sums
associated with the type 2 poly-Bernoulli functions of arbitrary indices, were introduced and were shown to satisfy a reciprocity relation. In this paper, as another generalization of the generalized Dedekind sums, we considered the poly-Dedekind sums
associated with the poly-Bernoulli functions of arbitrary indices, and derived a reciprocity relation for these poly-Dedekind sums.
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Acknowledgements
The authors would like to thank the reviewers for their suggestions that helped improve the original manuscript in its present form. Also, the authors thank Jangjeon Research Institute for Mathematical Sciences for the support of this research.
Funding
This research was funded by the National Natural Science Foundation of China (No. 11871317, 11926325, 11926321).
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Ma, Y., Kim, D.S., Lee, H. et al. Poly-Dedekind sums associated with poly-Bernoulli functions. J Inequal Appl 2020, 248 (2020). https://doi.org/10.1186/s13660-020-02513-7
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DOI: https://doi.org/10.1186/s13660-020-02513-7