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Fourier series of higher-order Bernoulli functions and their applications

Journal of Inequalities and Applications20172017:8

https://doi.org/10.1186/s13660-016-1282-y

Received: 3 November 2016

Accepted: 22 December 2016

Published: 5 January 2017

Abstract

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.

Keywords

Fourier seriesBernoulli polynomialsBernoulli functions

MSC

11B6842A16

1 Introduction

As is well known, Bernoulli polynomials are defined by the generating function
$$ \begin{aligned} \frac{t}{e^{t}-1}e^{xt} = \sum _{n=0}^{\infty}B_{n}(x) \frac{t^{n}}{n!}\quad (\mbox{see [1--23]}). \end{aligned} $$
(1.1)
When \(x=0\), \(B_{n}=B_{n}(0)\) are called Bernoulli numbers. From (1.1), we note that
$$ \begin{aligned} B_{n}(x) = \sum _{l=0}^{n} {n \choose l} B_{l} x^{n-l} \in\mathbb{Q}[x]\quad (n \geq0), \end{aligned} $$
(1.2)
with \(\deg B_{n}(x)=n\) (see [911]). By (1.1), we easily get
$$ \begin{aligned} (B+1)^{n} - B_{n} = \textstyle\begin{cases} 1,&\text{if } n=1,\\ 0,&\text{if }n>1, \end{cases}\displaystyle \quad\text{and}\quad B_{0}=1, \end{aligned} $$
(1.3)
with the usual convention about replacing \(B^{n}\) by \(B_{n}\) (see [9, 10]). From (1.2), we note that
$$ \begin{aligned}[b] \frac{dB_{n}(x)}{dx} &=\frac{d}{dx}\sum _{k=0}^{n} {n \choose k}B_{k}x^{n-k} = \sum_{k=0}^{n-1} {n \choose k}B_{k}(n-k) x^{n-k-1} \\ &=n \sum_{k=0}^{n-1} \frac{(n-1)!}{( n-k-1)!k!} B_{k} x^{n-k-1} = n \sum_{k=0}^{n-1} {n-1 \choose k}B_{k} x^{n-1-k} \\ &= nB_{n-1}(x) \quad (n \geq1)\ (\mbox{see [9--18]}). \end{aligned} $$
(1.4)
Thus, by (1.4), we get
$$ \begin{aligned} \int_{0}^{x} B_{n}(x)\,dx = \frac{1}{n+1} \bigl( B_{n+1}(x) - B_{n+1}(0) \bigr) \quad (n \geq0). \end{aligned} $$
(1.5)
For any real number x, we define
$$ \begin{aligned} \langle x \rangle= x-[x] \in[0,1), \end{aligned} $$
(1.6)
where \([x]\) is the integral part of x. Then \(B_{n}( \langle x \rangle)\) are functions defined on \((-\infty, \infty)\) and periodic with period 1, which are called Bernoulli functions. The Fourier series for \(B_{m}( \langle x \rangle)\) is given by
$$ \begin{aligned} B_{m}\bigl( \langle x \rangle \bigr) = -m! \sum_{\substack{n=-\infty\\n \neq0}}^{\infty}\frac{e^{2\pi inx}}{(2 \pi in)^{m}}, \end{aligned} $$
(1.7)
where \(m \geq1\) and \(x \notin\mathbb{Z}\) (see [1, 2, 8, 14, 22]). For a positive integer N, we have
$$ \begin{aligned} \sum_{k=0}^{N-1}B_{m} \biggl( \biggl\langle \frac{x+k}{N} \biggr\rangle \biggr) &=-m! \sum _{k=0}^{N-1} \sum _{\substack{n=-\infty\\n \neq0}}^{\infty}\frac{e^{2\pi in (\frac{x+k}{N} ) }}{(2 \pi in)^{m}} \\ &=-m!\sum_{\substack{n=-\infty\\n \neq0}}^{\infty}\frac{e^{2\pi in \frac {x}{N} }}{(2 \pi in)^{m}} \sum _{k=0}^{N-1} e^{2 \pi in \frac{k}{N}} \\ &=-m! N^{1-m} \sum_{\substack{l=-\infty\\l \neq0}}^{\infty}\frac {e^{2\pi il x }}{(2 \pi il)^{m}} = N^{1-m} B_{m}\bigl( \langle x \rangle \bigr) \quad (x \notin\mathbb{Z}). \end{aligned} $$
For \(r \in\mathbb{N}\), the higher-order Bernoulli polynomials are defined by the generating function
$$ \begin{aligned} \biggl( \frac{t}{e^{t}-1} \biggr)^{r} e^{xt} = \sum_{n=0}^{\infty}B_{n}^{(r)}(x) \frac{t^{n}}{n!}\quad(\mbox{see [1, 10, 11, 22]}). \end{aligned} $$
(1.8)
When \(x=0\), \(B_{n}^{(r)}= B_{n}^{(r)}(0)\) are called Bernoulli numbers of order r (see [1, 22]). Then \(B_{n}^{(r)}( \langle x \rangle)\) are functions defined on \((-\infty, \infty)\) and periodic with period 1, which are called Bernoulli functions of order r. In this paper, we study the Fourier series related to higher-order Bernoulli functions and give some new identities for the higher-order Bernoulli functions which are derived from the Fourier series of them.

2 Fourier series of higher-order Bernoulli functions and their applications

From (1.8), we note that
$$ \begin{aligned} B_{m}^{(r)}(x+1) = B_{m}^{(r)}(x) + m B_{m-1}^{(r-1)}(x) \quad (m \geq0). \end{aligned} $$
(2.1)
Indeed,
$$ \begin{aligned}[b] \sum_{m=0}^{\infty}B_{m}^{(r)}(x+1) \frac{t^{m}}{m!} &= \biggl( \frac {t}{e^{t}-1} \biggr)^{r} e^{(x+1)t}= \biggl( \frac{t}{e^{t}-1} \biggr)^{r} e^{xt}\bigl(e^{t} -1 +1\bigr) \\ &= \biggl( \frac{t}{e^{t}-1} \biggr)^{r-1} te^{xt}+ \biggl( \frac{t}{e^{t}-1} \biggr)^{r} e^{xt} \\ &=\sum_{m=0}^{\infty}B_{m}^{(r-1)}(x) \frac{t^{m+1}}{m!} + \sum_{m=0}^{\infty}B_{m}^{(r)}(x) \frac{t^{m}}{m!} \\ &= \sum_{m=0}^{\infty}\bigl( m B_{m-1}^{(r-1)}(x) + B_{m}^{(r)}(x) \bigr) \frac{t^{m}}{m!}. \end{aligned} $$
(2.2)
Let \(x=0\) in (2.1). Then we have
$$ \begin{aligned} B_{m}^{(r)}(1) = B_{m}^{(r)}(0) + mB_{m-1}^{(r-1)}(0)\quad ( m \geq0). \end{aligned} $$
(2.3)
Now, we assume that \(m \geq1\), \(r \geq2\). \(B_{m}^{(r)}( \langle x \rangle)\) is piecewise \(C^{\infty}\). Further, in view of (2.3), \(B_{m}^{(r)}( \langle x \rangle)\) is continuous for those \((r,m)\) with \(B_{m-1}^{(r-1)}(0) =0\), and is discontinuous with jump discontinuities at integers for those \((r,m)\) with \(B_{m-1}^{(r-1)}(0) \neq0\). The Fourier series of \(B_{m}^{(r)}( \langle x \rangle )\) is
$$ \begin{aligned} \sum_{n=-\infty}^{\infty}C_{n}^{(r,m)}e^{2 \pi inx}\quad (i = \sqrt{-1}), \end{aligned} $$
(2.4)
where
$$ \begin{aligned}[b] C_{n}^{(r,m)} &= \int_{0}^{1} B_{m}^{(r)}\bigl( \langle x \rangle\bigr) e^{-2 \pi inx} \,dx = \int_{0}^{1} B_{m}^{(r)}(x) e^{-2 \pi inx}\,dx \\ &= \biggl[ \frac{1}{m+1} B_{m+1}^{(r)}(x) e^{-2 \pi inx} \biggr]_{0}^{1} + \frac{2 \pi in}{m+1} \int_{0}^{1} B_{m+1}^{(r)}(x) e^{-2 \pi inx} \,dx \\ &= \frac{1}{m+1} \bigl( B_{m+1}^{(r)}(1) - B_{m+1}^{(r)}(0) \bigr) + \frac{2 \pi in}{m+1} C_{n}^{(r,m+1)} \\ &= B_{m}^{(r-1)}(0) + \frac{2 \pi in}{m+1} C_{n}^{(r,m+1)}. \end{aligned} $$
(2.5)
Replacing m by \(m-1\) in (2.5), we get
$$ \begin{aligned} C_{n}^{(r,m-1)} = B_{m-1}^{(r-1)}(0) + \frac{2 \pi in}{m} C_{n}^{(r,m)}. \end{aligned} $$
(2.6)
Case 1. Let \(n \neq0\). Then we have
$$\begin{aligned} C_{n}^{(r,m)}&= \frac{m}{2 \pi in } C_{n}^{(r,m-1)}- \frac{m}{2\pi in}B_{m-1}^{(r-1)} \\ &= \frac{m}{2\pi in} \biggl( \frac{m-1}{2\pi in} C_{n}^{(r,m-2)}- \frac {m-1}{2\pi in} B_{m-2}^{(r-1)} \biggr) - \frac{m}{2\pi in}B_{m-1}^{(r-1)} \\ &= \frac{m(m-1)}{(2\pi in)^{2}}C_{n}^{(r,m-2)} - \frac{m(m-1)}{(2\pi in)^{2}}B_{m-2}^{(r-1)} - \frac{m}{2\pi in}B_{m-1}^{(r-1)} \\ &=\frac{m(m-1)}{(2\pi in)^{2}} \biggl\{ \frac{m-2}{2\pi in} C_{n}^{(r,m-3)}- \frac{m-2}{2\pi in} B_{m-3}^{(r-1)} \biggr\} - \frac {m(m-1)}{(2\pi in)^{2}}B_{m-2}^{(r-1)} - \frac{m}{2\pi in}B_{m-1}^{(r-1)} \\ &= \frac{m(m-1)(m-2)}{(2\pi in)^{3}} C_{n}^{(r,m-3)} - \frac {m(m-1)(m-2)}{(2\pi in)^{3}} B_{m-3}^{(r-1)} \\ &\quad {} - \frac{m(m-1)}{(2\pi in)^{2}}B_{m-2}^{(r-1)} - \frac{m}{2\pi in}B_{m-1}^{(r-1)} \\ &= \cdots \\ &= \frac{m(m-1)(m-2)\cdots2}{(2\pi in)^{m-1}} C_{n}^{(r,1)} - \sum _{k=1}^{m-1} \frac{(m)_{k}}{(2\pi in)^{k}} B_{m-k}^{(r-1)}, \end{aligned}$$
(2.7)
where \((x)_{n} = x(x-1)\cdots(x-n+1)\), for \(n \geq1\), and \((x)_{0}=1\). Now, we observe that
$$ \begin{aligned}[b] C_{n}^{(r,1)} &= \int_{0}^{1} B_{1}^{(r)}(x) e^{-2 \pi inx}\,dx = \int_{0}^{1} \bigl(x+B_{1}^{(r)} \bigr) e^{-2 \pi inx} \,dx \\ &= \int_{0}^{1} xe^{-2 \pi inx}\,dx + B_{1}^{(r)} \int_{0}^{1} e^{-2 \pi inx}\,dx \\ &= - \frac{1}{2 \pi in} \bigl[ x e^{-2 \pi inx} \bigr]_{0}^{1} + \frac {1}{2 \pi in} \int_{0}^{1} e^{-2 \pi inx} \,dx \\ &= - \frac{1}{2 \pi in}. \end{aligned} $$
(2.8)
From (2.7) and (2.8), we can derive equation (2.9):
$$ \begin{aligned}[b] C_{n}^{(r,m)} &= - \frac{m!}{(2 \pi in)^{m}} - \sum_{k=1}^{m-1} \frac {(m)_{k}}{(2 \pi in)^{k}} B_{m-k}^{(r-1)} \\ &= - \sum_{k=1}^{m} \frac {(m)_{k}}{(2 \pi in)^{k}}B_{m-k}^{(r-1)}. \end{aligned} $$
(2.9)
Case 2. Let \(n=0\). Then we have
$$ \begin{aligned}[b] C_{0}^{(r,m)} &= \int_{0}^{1} B_{m}^{(r)}\bigl( \langle x \rangle\bigr) \,dx = \int_{0}^{1} B_{m}^{(r)}(x) \,dx = \frac{1}{m+1} \bigl[ B_{m+1}^{(r)}(x) \bigr]_{0}^{1} \\ &= \frac{1}{m+1} \bigl( B_{m+1}^{(r)}(1) - B_{m+1}^{(r)}(0) \bigr) = B_{m}^{(r-1)}. \end{aligned} $$
(2.10)
Before proceeding, we recall the following equations:
$$ \begin{aligned} B_{m}\bigl( \langle x \rangle \bigr) = -m! \sum_{\substack{n=-\infty\\ n \neq0}}^{\infty}\frac{e^{2 \pi inx}}{(2 \pi in)^{m}}, \quad (m \geq2) \ (\mbox{see [1]}), \end{aligned} $$
(2.11)
and
$$ \begin{aligned} -\sum_{\substack{n=-\infty\\ n \neq0}}^{\infty}\frac{e^{2 \pi inx}}{2 \pi in} = \textstyle\begin{cases} B_{1}( \langle x \rangle),& \text{for } x \notin\mathbb {Z},\\ 0,& \text{for } x \in\mathbb{Z},\ (\mbox{see [1, 22]}). \end{cases}\displaystyle \end{aligned} $$
(2.12)
The series in (2.11) converges uniformly, while that in (2.12) converges pointwise. Assume first that \(B_{m-1}^{(r-1)}(0) =0\). Then we have \(B_{m}^{(r)}(1) = B_{m}^{(r)}(0)\), and \(m \geq2\). As \(B_{m}^{(r)}( \langle x \rangle)\) is piecewise \(C^{\infty}\) and continuous, the Fourier series of \(B_{m}^{(r)}( \langle x \rangle)\) converges uniformly to \(B_{m}^{(r)}( \langle x \rangle)\), and
$$ \begin{aligned}[b] B_{m}^{(r)}\bigl( \langle x \rangle\bigr) &= \sum_{n=-\infty}^{\infty}C_{n}^{(r,m)}e^{2 \pi inx} \\ &= B_{m}^{(r-1)} - \sum_{\substack{n=-\infty\\ n \neq0}}^{\infty}\Biggl(\sum_{k=1}^{m} \frac{(m)_{k}}{(2 \pi in)^{k}}B_{m-k}^{(r-1)} \Biggr) e^{2 \pi inx} \\ &= B_{m}^{(r-1)} + \sum_{k=1}^{m} \frac{(m)_{k}}{k!} B_{m-k}^{(r-1)} \cdot \Biggl( -k! \sum _{\substack{n=-\infty\\ n \neq0}}^{\infty}\frac{e^{2 \pi inx}}{(2 \pi in)^{k}} \Biggr) \\ &= B_{m}^{(r-1)} + \sum_{k=2}^{m} {m \choose k} B_{m-k}^{(r-1)} B_{k} \bigl( \langle x \rangle\bigr) \\ &\quad{}+ {m \choose 1} B_{m-1}^{(r-1)} \times \textstyle\begin{cases} B_{1}( \langle x \rangle),& \text{for } x \notin\mathbb {Z},\\ 0,& \text{for } x \in\mathbb{Z} \end{cases}\displaystyle \\ &= \textstyle\begin{cases} \sum_{k=0}^{m} {m \choose k} B_{m-k}^{(r-1)}B_{k}( \langle x \rangle) & \text{for } x \notin\mathbb{Z},\\ \sum_{\substack{k=0\\k \neq1}}^{m} {m \choose k} B_{m-k}^{(r-1)}B_{k}( \langle x \rangle) & \text{for } x \in\mathbb{Z}. \end{cases}\displaystyle \end{aligned} $$
(2.13)
Note that (2.13) holds whether \(B_{m-1}^{(r-1)}(0)=0\) or not. However, if \(B_{m-1}^{(r-1)}(0)=0\), then
$$ \begin{aligned} B_{m}^{(r)}\bigl( \langle x \rangle \bigr) = \sum_{\substack{k=0 \\ k \neq 1}}^{m} {m \choose k} B_{m-k}^{(r-1)} B_{k} \bigl( \langle x \rangle\bigr), \quad \text{for all } x \in(-\infty, \infty). \end{aligned} $$
Therefore, we obtain the following theorem.

Theorem 2.1

Let \(m \geq2\), \(r \geq2\). Assume that \(B_{m-1}^{(r-1)}(0)=0\).
  1. (a)
    \(B_{m}^{(r)}( \langle x \rangle)\) has the Fourier series expansion
    $$ \begin{aligned} B_{m}^{(r)}\bigl( \langle x \rangle \bigr) = B_{m}^{(r-1)}(0) - \sum_{\substack{n=-\infty\\ n \neq0}}^{\infty}\Biggl(\sum_{k=1}^{m} \frac {(m)_{k}}{(2 \pi in)^{k}}B_{m-k}^{(r-1)} \Biggr) e^{2 \pi inx}, \end{aligned} $$
    for \(x \in(-\infty, \infty)\). Here the convergence is uniform.
     
  2. (b)

    \(B_{m}^{(r)}( \langle x \rangle) = \sum_{\substack{k=0 \\ k \neq1}}^{m} {m \choose k} B_{m-k}^{(r-1)} B_{k} ( \langle x \rangle)\), for all \(x \in(-\infty, \infty)\), where \(B_{k}( \langle x \rangle)\) is the Bernoulli function.

     

Assume next that \(B_{m-1}^{(r-1)}(0) \neq0\). Then \(B_{m}^{(r)}(1) \neq B_{m}^{(r)}(0)\), and hence \(B_{m}^{(r)}( \langle x \rangle)\) is piecewise \(C^{\infty}\) and discontinuous with jump discontinuities at integers Thus the Fourier series of \(B_{m}^{(r)}( \langle x \rangle)\) converges pointwise to \(B_{m}^{(r)}( \langle x \rangle)\), for \(x \notin\mathbb{Z}\), and converges to \(\frac{1}{2} ( B_{m}^{(r)}(0) + B_{m}^{(r)}(1) ) = B_{m}^{(r)}(0) + \frac{m}{2} B_{m-1}^{(r-1)}(0)\), for \(x \in\mathbb{Z}\). Thus we obtain the following theorem.

Theorem 2.2

Let \(m \geq1\), \(r \geq2\), Assume that \(B_{m-1}^{(r-1)}(0) \neq0\).
$$ \begin{aligned} (\mathrm{a}) \quad B_{m}^{(r-1)}(0) - \sum_{\substack{n=-\infty\\ n \neq0}}^{\infty}\Biggl(\sum _{k=1}^{m} \frac{(m)_{k}}{(2 \pi in)^{k}}B_{m-k}^{(r-1)} \Biggr) e^{2 \pi inx} = \textstyle\begin{cases} B_{m}^{(r)}( \langle x \rangle),& \textit{for } x \notin \mathbb{Z},\\ B_{m}^{(r)} + \frac{m}{2} B_{m-1}^{(r-1)},& \textit{for } x \in\mathbb{Z}. \end{cases}\displaystyle \end{aligned} $$
Here the convergence is pointwise,
$$ \begin{aligned} (\mathrm{b}) \quad \sum_{k=0}^{m} {m \choose k} B_{m-k}^{(r-1)} B_{k} \bigl( \langle x \rangle\bigr) = B_{m}^{(r)}(x), \quad \textit{for } x \notin\mathbb{Z}, \end{aligned} $$
and
$$ \begin{aligned} \sum_{\substack{k=0 \\ k \neq1}}^{m} {m \choose k} B_{m-k}^{(r-1)} B_{k} \bigl( \langle x \rangle\bigr) = B_{m}^{(r)} + \frac{m}{2} B_{m-1}^{(r-1)}, \quad \textit{for } x \in\mathbb{Z}, \end{aligned} $$
where \(B_{k}( \langle x \rangle)\) is the Bernoulli function.

Remark

Let \(\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}\), \(( \operatorname{Re}(s)>1 )\). From (1.7), we note that, for \(m \geq1\),
$$ \begin{aligned} B_{2m} &= -(2m)! \sum _{\substack{n=-\infty\\n \neq0}}^{\infty}\frac {(-1)^{m}}{(2 \pi n)^{2m}} \\ &= - \frac{(2m)!}{(2\pi)^{2m}} 2 \sum_{n=1}^{\infty}\frac {(-1)^{m}}{n^{2m}} = (-1)^{m+1} \frac{2(2m)!}{(2\pi)^{2m}} \zeta(2m). \end{aligned} $$

3 Results and discussion

In this paper, we studied the Fourier series expansion of the higher-order Bernoulli functions \(B_{m}^{(r)}( \langle x \rangle )\) which are obtained by extending by periodicity of period 1 the higher-order Bernoulli polynomials \(B_{m}^{(r)}(x)\) on \([0, 1)\). As it turns out, the Fourier series of \(B_{m}^{(r)}( \langle x \rangle)\) converges uniformly to \(B_{m}^{(r)}( \langle x \rangle)\), if \(B_{m-1}^{(r-1)}(0)=0\), and converges pointwise to \(B_{m}^{(r)}( \langle x \rangle)\) for \(x\notin \Bbb {Z}\) and converges to \(B_{m}^{(r)}+\frac{m}{2}B_{m-1}^{(r-1)}\) for \(x\in\Bbb {Z}\), if \(B_{m-1}^{(r-1)}(0)\neq0\). Here the Fourier series of the higher-order Bernoulli functions \(B_{m}^{(r)}( \langle x \rangle)\) are explicitly determined. In addition, in each case the Fourier series of the higher-order Bernoulli functions \(B_{m}^{(r)}( \langle x \rangle)\) are expressed in terms of Bernoulli functions which are obtained by extending by periodicity of period 1 the ordinary Bernoulli polynomials \(B_{m}(x)\) on \([0, 1)\). The Fourier series expansion of the Bernoulli functions are useful in computing the special values of the Dirichlet L-functions. For details, one is referred to [24].

It is expected that the Fourier series of the higher-order Bernoulli functions will find some applications in connections with a certain generalization of Dirichlet L-functions and higher-order generalized Bernoulli numbers.

4 Conclusion

In this paper, we considered the Fourier series expansion of the higher-order Bernoulli functions \(B_{m}^{(r)}( \langle x \rangle )\) which are obtained by extending by periodicity of period 1 the higher-order Bernoulli polynomials \(B_{m}^{(r)}(x)\) on \([0, 1)\). The Fourier series are explicitly determined. Depending on whether \(B_{m-1}^{(r-1)}(0)\) is zero or not, the Fourier series of \(B_{m}^{(r)}( \langle x \rangle)\) converges uniformly to \(B_{m}^{(r)}( \langle x \rangle)\) or converges pointwise to \(B_{m}^{(r)}( \langle x \rangle)\) for \(x\notin\Bbb {Z}\) and converges to \(B_{m}^{(r)}+\frac {m}{2}B_{m-1}^{(r-1)}\) for \(x\in\Bbb {Z}\). In addition, the Fourier series of the higher-order Bernoulli functions \(B_{m}^{(r)}( \langle x \rangle)\) are expressed in terms of Bernoulli functions \(B_{k}( \langle x \rangle)\). Thus we established the relations between higher-order Bernoulli functions and Bernoulli functions. Just as the Fourier series expansion of the Bernoulli functions are useful in computing the special values of Dirichlet L-functions, we would like to see some applications to a certain generalization of Dirichlet L-functions and higher-order generalized Bernoulli numbers in near future.

Declarations

Acknowledgements

This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea
(2)
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, China
(3)
Department of Mathematics, Sogang University, Seoul, Republic of Korea
(4)
Department of Mathematics Education, Kyungpook National University, Taegu, Republic of Korea
(5)
Hanrimwon, Kwangwoon University, Seoul, Republic of Korea
(6)
School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia

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