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 [9–11]). 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.