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Identities on poly-Dedekind sums
Advances in Difference Equations volume 2020, Article number: 563 (2020)
Abstract
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.
1 Introduction
To give concise definition of the Dedekind sums, we introduce the notation
where \([x]\) denotes the greatest integer not exceeding x.
It is well known that the Dedekind sums are defined by
where h is any integer.
From (2) we note that
As is well known, the Bernoulli polynomials are given by
When \(x=0\), \(B_{n}=B_{n}(0)\) (\(n\ge 0\)) are called the Bernoulli numbers.
From (4) we note that
with the usual convention about replacing \(B^{n}\) by \(B_{n}\).
We observe that
Thus by (6) we get
Recently, Kim and Kim [5, 9] considered the polyexponential function of index k given by
Note that \(\operatorname{Ei}_{1}(x)=e^{x}-1\).
In [5] the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k as
When \(x=0\), \(B_{n}^{(k)}=B_{n}^{(k)}(0)\) (\(n\ge 0\)) are called the type 2 poly-Bernoulli numbers of index k. Note that \(B_{n}^{(1)}(x)=B_{n}(x)\) are the Bernoulli polynomials.
The fractional part of x is denoted by
The Bernoulli functions are defined by
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:
-
(a)
\(\sum_{i=0}^{d-1} B_{n} (\frac{x+i}{d} )=d^{1-n}B_{n}(x)\),
-
(b)
\(\sum_{i=0}^{d-1} \overline{B}_{n} (\frac{x+i}{d} )=d^{1-n} \overline{B}_{n}(x)\), and
-
(c)
\(\sum_{i=0}^{d-1} B_{n} (\frac{\langle x \rangle +i}{d} )= \sum_{i=0}^{d-1} \overline{B}_{n} (\frac{x+i}{d} )\) for all real x.
Dedekind showed that the quantity \(S(h,m) = \sum_{\mu =1}^{m-1}\frac{\mu }{m}\overline{B}_{1} ( \frac{h\mu }{m} )\) 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 \(B_{p}^{(k)}(x)\) are the type 2 poly-Bernoulli polynomials of index k (see (9)), and \(\overline{B}_{p}^{(k)}(x)=B_{p}^{(k)}(\langle x \rangle )\) are the type 2 poly-Bernoulli functions of index k. Note that \(S_{p}^{(1)}(h,m)=S_{p}(h,m)\). 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.
2 On type 2 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 \(n\ge 1\), we have
By Theorem 2 we get
where \(\delta _{n,k}\) 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.
Theorem 3
For \(s,p\in \mathbb{N}\), we have
Now we observe that
Therefore by Theorem 3 and (21) we obtain the following corollary.
Corollary 4
For \(s,p\in \mathbb{N}\), we have
From (16) we have
On the other hand, by (15) we get
Therefore by (22) and (23) we obtain the following theorem.
Theorem 5
For \(p\in \mathbb{N}\), we have
3 Poly-Dedekind sums
Apostol considered the generalized Dedekind sums given by
where \(\overline{B}_{p}(h\mu /m)=B_{p} (\langle h\mu /m\rangle )\).
Note that, for any relatively prime positive integers h, m, we have
In this section, we consider the poly-Dedekind sums given by
where \(h,m,p\in \mathbb{N}\), \(k\in \mathbb{Z}\), and \(\overline{B}_{p}^{(k)}(x)=B_{p}^{(k)}(\langle x \rangle )\) are the type 2 poly-Bernoulli functions of index k.
Note that
Assume now that \(h=1\). Then we have
From (5) we have
Now we assume that \(p\ge 3\) is an odd positive integer, so that \(B_{p}=0\). Then we have
Therefore by (29) we obtain the following proposition.
Proposition 6
Let \(p\ge 3\) be an odd positive integer. Then we have
We still assume that \(p\ge 3\) 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 \(\binom{p}{i-2}\frac{p+1}{i}=\frac{1}{p+2}\binom{p+2}{i}(i-1)\) for \(i \ge 1\) and that \(B_{1}^{(k)}(1)-B_{1}^{(k)}=1\) by Theorem 2. Then from (30) we see that
Therefore by (31) we obtain the following theorem.
Theorem 7
For \(m\in \mathbb{N}\) and any odd positive integer \(p \ge 3\), we have
Now we employ the notation
Assume that h, m are relatively prime positive integers. Then we see that
Now, as the index μ ranges over the values \(\mu =0,1,2,\ldots,m-1\), the product hμ ranges over a complete residue system modulo m, and due to the periodicity of \(\overline{B}_{1}(x)\), the term \(\overline{B}_{1}(h\mu/m)\) may be replaced by \(\overline{B}_{1}(\mu /m)\) without altering the sum over μ. Thus the sum (32) is equal to
where we used the fact (a) in Lemma 1.
Therefore we obtain the following theorem.
Theorem 8
For \(m,n,h\in \mathbb{N}\) with \((h,m)=1\) and any positive odd integer \(p\ge 3\), we have
Now we observe that
where d is a positive integer.
Therefore by comparing the coefficients on both sides of (34) we obtain the following theorem.
Theorem 9
For \(k\in \mathbb{Z}\), \(d\in \mathbb{N}\), and \(n\ge 0\), we have
From (25), using Theorem 9 and (c) in Lemma 1, we see that
Therefore we obtain the following reciprocity relation.
Theorem 10
For \(m,h,p\in \mathbb{N}\) and \(k\in \mathbb{Z}\), we have
In the case \(k=1\), we obtain the following reciprocity relation for the generalized Dedekind sum defined by Apostol.
Corollary 11
For \(m,h,p\in \mathbb{N}\), we have
4 Conclusion
The Dedekind sums are defined by
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
In [5] the type 2 poly-Bernoulli polynomials of index k are defined in terms of the polyexponential function of index k by
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.
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The authors thank Jangjeon Institute for Mathematical Science for the support of this research.
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TK and DSK conceived of the framework and structured the whole paper; TK and DSK wrote the paper; L-CJ and HL checked the results of the paper; DSK and TK completed the revision of the paper. All authors have read and agreed to the published version of the manuscript.
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Kim, T., Kim, D.S., Lee, H. et al. Identities on poly-Dedekind sums. Adv Differ Equ 2020, 563 (2020). https://doi.org/10.1186/s13662-020-03024-x
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DOI: https://doi.org/10.1186/s13662-020-03024-x