Fourier series of sums of products of ordered Bell and poly-Bernoulli functions

Abstract

In this paper, we study three types of sums of products of ordered Bell and poly-Bernoulli functions and derive their Fourier series expansion. In addition, we express those functions in terms of Bernoulli functions.

Introduction

The ordered Bell polynomials $b_m(x)$ are defined by the generating function

$$\frac{1}{2-e^t}{e}^{xt}=\sum _{m=0}^{\mathrm{\infty }}{b}_m(x)\frac{t^m}{m!}.$$ 1

Thus they form an Appell sequence. For $x=0$, $b_m=b_m(0)$, $(m\ge 0)$ are called ordered Bell numbers which have been studied in various counting problems in number theory and enumerative combinatorics (see [1, 4, 5, 16, 17, 19]). The ordered Bell numbers are all positive integers, as we can see, for example, from

$$b_m=\sum _{n=0}^{m}n!S_2(m,n)=\sum _{n=0}^{\mathrm{\infty }}\frac{n^m}{2^{n+1}}(m\ge 0).$$

The first few ordered Bell polynomials are as follows:

$$\begin{array}{c}{b}_0(x)=1,b_1(x)=x+1,b_2(x)=x^2+2x+3,\hfill \\ b_3(x)=x^3+3x^2+9x+13,b_4(x)=x^4+4x^3+18x^2+52x+75,\hfill \\ b_5(x)=x^5+5x^4+30x^3+130x^2+375x+541.\hfill \end{array}$$

From (1), we can derive

$$\begin{array}{c}\frac{d}{dx}{b}_m(x)=m{b}_{m-1}(x)(m\ge 1),\hfill \\ -b_m(x+1)+2b_m(x)=x^m(m\ge 0).\hfill \end{array}$$

From these, in turn, we have

$$\begin{array}{c}-b_m(1)+2b_m={\delta }_{m,0}(m\ge 0),\hfill \\ {\int }_0^1{b}_m(x)dx=\frac{1}{m+1}(b_{m+1}(1)-m_{m+1})\hfill \\ \phantom{in{t}_0^1{b}_m(x)dx}=\frac{1}{m+1}{b}_{m+1}.\hfill \end{array}$$

For any integer r, the poly-Bernoulli polynomials of index r ${\mathbb{B}}_m^{(r)}(x)$ are given by the generating function (see [2, 3, 7–10, 12, 13, 20])

$$\frac{L{i}_r(1-e^{-t})}{e^t-1}{e}^{xt}=\sum _{m=0}^{\mathrm{\infty }}{\mathbb{B}}_m^{(r)}(x)\frac{t^m}{m!},$$

where $L{i}_r(x)={\sum }_{m=1}^{\mathrm{\infty }}\frac{x^m}{m^r}$ is the polylogarithmic function for $r\ge 1$ and a rational function for $r\le 0$.

We note here that

$$\frac{d}{dx}(L{i}_{r+1}(x))=\frac{1}{x}L{i}_r(x).$$

Also, we need the following for later use.

$$\begin{array}{c}\frac{d}{dx}{\mathbb{B}}_m^{(r)}(x)=m{\mathbb{B}}_{m-1}^{(r)}(x)(m\ge 1),\hfill \\ {\mathbb{B}}_m^{(1)}(x)=B_m(x),{\mathbb{B}}_0^{(r)}(x)=1,{\mathbb{B}}_m^{(0)}(x)=x^m,\hfill \\ {\mathbb{B}}_m^{(0)}={\delta }_{m,0},{\mathbb{B}}_m^{(r+1)}(1)-{\mathbb{B}}_m^{(r+1)}(0)={\mathbb{B}}_{m-1}^{(r)}(m\ge 1),\hfill \\ {\int }_0^1{\mathbb{B}}_m^{(r)}(x)dx=\frac{1}{m+1}({\mathbb{B}}_{m+1}^{(r)}(1)-{\mathbb{B}}_{m+1}^{(r)}(0))\hfill \\ \phantom{{\int }_0^1{\mathbb{B}}_m^{(r)}(x)dx}=\frac{1}{m+1}{\mathbb{B}}_m^{(r-1)}.\hfill \end{array}$$

Here the Bernoulli polynomials $B_m(x)$ are given by the generating function

$$\frac{t}{e^t-1}{e}^{xt}=\sum _{m=0}^{\mathrm{\infty }}{B}_m(x)\frac{t^m}{m!}.$$

For any real number x, we let

$$\langle x\rangle =x-\lfloor x\rfloor \in [0,1)$$

denote the fractional part of x.

Finally, we recall the following facts about Bernoulli functions $B_m(\langle x\rangle )$:

1. for $m\ge 2$,

   $$B_m(\langle x\rangle )=-m!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^m};$$
2. for $m=1$,

   $$-\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{2\pi in}=\{\begin{array}{cc}{B}_1(\langle x\rangle ),\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0,\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}$$

Here we will study three types of sums of products of ordered Bell and poly-Bernoulli functions and derive their Fourier series expansion. In addition, we will express those functions in terms of Bernoulli functions.

1. ${\alpha }_m(\langle x\rangle )={\sum }_{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),(m\ge 1)$;

2. ${\beta }_m(\langle x\rangle )={\sum }_{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),(m\ge 1)$;

3. ${\gamma }_m(\langle x\rangle )={\sum }_{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),(m\ge 2)$.

For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [18, 21]).

As to ${\gamma }_m(\langle x\rangle )$, we note that the next polynomial identity follows immediately from Theorems 4.1 and 4.2, which is in turn derived from the Fourier series expansion of ${\gamma }_m(\langle x\rangle )$:

$$\begin{array}{c}\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(x){\mathbb{B}}_{m-k}^{(r+1)}(x)\hfill \\ =\frac{1}{m}\sum _{s=0}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(x),\hfill \end{array}$$

where $H_l={\sum }_{j=1}^l\frac{1}{j}$ are the harmonic numbers and

$${\mathrm{\Lambda }}_l=\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k-1}^{(r)},$$

with ${\mathrm{\Lambda }}_1=0$.

The polynomial identities can be derived also for the functions ${\alpha }_m(\langle x\rangle )$ and ${\beta }_m(\langle x\rangle )$ from Theorems 2.1 and 2.2, and Theorems 3.1 and 3.2, respectively. We refer the reader to [6, 11, 14, 15] for the recent papers on related works.

Fourier series of functions of the first type

Let

$${\alpha }_m(x)=\sum _{k=0}^m{b}_k(x){\mathbb{B}}_{m-k}^{(r+1)}(x),$$

where $r,m$ are integers with $m\ge 1$. Then we will study the function

$${\alpha }_m(\langle x\rangle )=\sum _{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )(m\ge 1),$$

defined on $\mathbb{R}$ which is periodic with period 1.

The Fourier series of ${\alpha }_m(\langle x\rangle )$ is

$$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{A}_n^{(m)}{e}^{2\pi inx},$$

where

$$\begin{array}{rcl}{A}_n^{(m)}& =& {\int }_0^1{\alpha }_m(\langle x\rangle )e^{-2\pi inx}dx\\ & =& {\int }_0^1{\alpha }_m(x)e^{-2\pi inx}dx.\end{array}$$

Before proceeding further, we observe the following

$$\begin{array}{rcl}{\alpha }_m^{\prime }(x)& =& \sum _{k=0}^m\{k{b}_{k-1}(x){\mathbb{B}}_{m-k}^{(r+1)}(x)+(m-k)b_k{\mathbb{B}}_{m-k-1}^{(r+1)}(x)\}\\ & =& \sum _{k=1}^{m}k{b}_{k-1}(x){\mathbb{B}}_{m-k}^{(r+1)}(x)+\sum _{k=0}^{m-1}(m-k)b_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& \sum _{k=0}^{m-1}(k+1)b_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)+\sum _{k=0}^{m-1}(m-k)b_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& (m+1){\alpha }_{m-1}(x).\end{array}$$

Thus ${(\frac{{\alpha }_{m+1}(x)}{m+2})}^{\prime }={\alpha }_m(x)$, and so ${\int }_0^1{\alpha }_m(x)dx=\frac{1}{m+2}({\alpha }_{m+1}(1)-{\alpha }_{m+1}(0))$. For $m\ge 1$, we put

$$\begin{array}{rcl}{\mathrm{\Delta }}_m& =& {\alpha }_m(1)-{\alpha }_m(0)\\ & =& \sum _{k=0}^m{b}_k(1){\mathbb{B}}_{m-k}^{(r+1)}(1)-\sum _{k=0}^m{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& \sum _{k=0}^{m-1}(2b_k-{\delta }_{k,0})({\mathbb{B}}_{m-k}^{(r+1)}+{\mathbb{B}}_{m-k-1}^{(r)})+2b_m-{\delta }_{m,0}-\sum _{k=0}^m{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& 2\sum _{k=0}^{m-1}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=0}^{m-1}{b}_k{\mathbb{B}}_{m-k-1}^{(r)}-{\mathbb{B}}_m^{(r+1)}-{\mathbb{B}}_{m-1}^{(r)}+b_m-\sum _{k=0}^{m-1}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& \sum _{k=1}^m{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=0}^{m-1}{b}_k{\mathbb{B}}_{m-k-1}^{(r)}-{\mathbb{B}}_{m-1}^{(r)}.\end{array}$$

Thus, ${\alpha }_m(0)={\alpha }_m(1)⟺{\mathrm{\Delta }}_m=0$ and ${\int }_0^1{\alpha }_m(x)dx=\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}$.

Now, we want to determine the Fourier coefficients $A_n^{(m)}$.

Case 1: $n\ne 0$.

$$\begin{array}{rcl}{A}_n^{(m)}& =& {\int }_0^1{\alpha }_m(x)e^{-2\pi inx}dx\\ & =& -\frac{1}{2\pi in}{[{\alpha }_m(x)e^{-2\pi inx}]}_0^1+\frac{1}{2\pi in}{\int }_0^1{\alpha }_m^{\prime }(x)e^{-2\pi inx}dx\\ & =& \frac{m+1}{2\pi in}{\int }_0^1{\alpha }_{m-1}(x)e^{-2\pi inx}dx-\frac{1}{2\pi in}({\alpha }_m(1)-{\alpha }_m(0))\\ & =& \frac{m+1}{2\pi in}{A}_n^{(m-1)}-\frac{1}{2\pi in}{\mathrm{\Delta }}_m\\ & =& \frac{m+1}{2\pi in}(\frac{m}{2\pi in}{A}_n^{(m-2)}-\frac{1}{2\pi in}{\mathrm{\Delta }}_{m-1})-\frac{1}{2\pi in}{\mathrm{\Delta }}_m\\ & =& \frac{{(m+1)}_2}{{(2\pi in)}^2}{A}_n^{(m-2)}-\sum _{j=1}^2\frac{{(m+1)}_{j-1}}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1}\\ & =& \cdots \\ & =& \frac{{(m+1)}_m}{{(2\pi in)}^m}{A}_n^{(0)}-\sum _{j=1}^m\frac{{(m+1)}_{j-1}}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1}\\ & =& -\frac{1}{m+2}\sum _{j=1}^m\frac{{(m+2)}_j}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1},\end{array}$$

where we note that $A_n^{(0)}={\int }_0^1{e}^{-2\pi inx}dx=0$.

Case 2: $n=0$.

$$A_0^{(m)}={\int }_0^1{\alpha }_m(x)dx=\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}.$$

${\alpha }_m(\langle x\rangle )$, $(m\ge 1)$ is piecewise $C^{\mathrm{\infty }}$. Moreover, ${\alpha }_m(\langle x\rangle )$ is continuous for those positive integers m with ${\mathrm{\Delta }}_m=0$ and discontinuous with jump discontinuities at integers for those positive integers m with ${\mathrm{\Delta }}_m\ne 0$.

Assume first that m is a positive integer with ${\mathrm{\Delta }}_m=0$. Then ${\alpha }_m(0)={\alpha }_m(1)$. Hence ${\alpha }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and continuous. Thus the Fourier series of ${\alpha }_m(\langle x\rangle )$ converges uniformly to ${\alpha }_m(\langle x\rangle )$, and

$$\begin{array}{rcl}{\alpha }_m(\langle x\rangle )& =& \frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\frac{1}{m+2}\sum _{j=1}^m\frac{{(m+2)}_j}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1})e^{2\pi inx}\\ & =& \frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\frac{1}{m+2}\sum _{j=1}^m\left(\genfrac{}{}{0ex}{}{m+2}{j}\right){\mathrm{\Delta }}_{m-j+1}(-j!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^j})\\ & =& \frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\frac{1}{m+2}\sum _{j=2}^m\left(\genfrac{}{}{0ex}{}{m+2}{j}\right){\mathrm{\Delta }}_{m-j+1}{B}_j(\langle x\rangle )\\ & & +{\mathrm{\Delta }}_m\times \{\begin{array}{cc}{B}_1(\langle x\rangle ),\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0,\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}\end{array}$$

Now, we can state our first theorem.

Theorem 2.1

For each positive integer l, we let

$${\mathrm{\Delta }}_l=\sum _{k=1}^l{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=0}^{l-1}{b}_k{\mathbb{B}}_{l-k-1}^{(r)}-{\mathbb{B}}_{l-1}^{(r)}.$$

Assume that ${\mathrm{\Delta }}_m=0$ for a positive integer m. Then we have the following

1. ${\sum }_{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )$ has the Fourier series expansion

   $$\begin{array}{c}\sum _{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )\hfill \\ =\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\frac{1}{m+2}\sum _{j=1}^m\frac{{(m+2)}_j}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1})e^{2\pi inx},\hfill \end{array}$$

   for all $x\in \mathbb{R}$, where the convergence is uniform.
2. $$\sum _{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )=\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\frac{1}{m+2}\sum _{j=2}^m\left(\genfrac{}{}{0ex}{}{m+2}{j}\right){\mathrm{\Delta }}_{m-j+1}{B}_j(\langle x\rangle ),$$

   for all $x\in \mathbb{R}$, where $B_j(\langle x\rangle )$ is the Bernoulli function.

Assume next that ${\mathrm{\Delta }}_m\ne 0$ for a positive integer m. Then ${\alpha }_m(0)\ne {\alpha }_m(1)$. So ${\alpha }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and discontinuous with jump discontinuities at integers. The Fourier series of ${\alpha }_m(\langle x\rangle )$ converges pointwise to ${\alpha }_m(\langle x\rangle )$, for $x\notin \mathbb{Z}$, and converges to

$$\frac{1}{2}({\alpha }_m(0)+{\alpha }_m(1))={\alpha }_m(0)+\frac{1}{2}{\mathrm{\Delta }}_m,$$

for $x\in \mathbb{Z}$.

Now, we can state our second theorem.

Theorem 2.2

For each positive integer l, we let

$${\mathrm{\Delta }}_l=\sum _{k=1}^l{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=0}^{l-1}{b}_k{\mathbb{B}}_{l-k-1}^{(r)}-{\mathbb{B}}_{l-1}^{(r)}.$$

Assume that ${\mathrm{\Delta }}_m\ne 0$ for a positive integer m. Then we have the following.

1. $$\begin{array}{c}\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\frac{1}{m+2}\sum _{j=1}^m\frac{{(m+2)}_j}{{(2\pi in)}^j}{\mathrm{\Delta }}_{m-j+1})e^{2\pi inx}\hfill \\ =\{\begin{array}{cc}{\sum }_{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\hfill & \mathit{\text{for }}x\notin \mathbb{Z},\hfill \\ {\sum }_{k=0}^m{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Delta }}_m,\hfill & \mathit{\text{for }}x\in \mathbb{Z};\hfill \end{array}\hfill \end{array}$$
2. $$\begin{array}{c}\frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\frac{1}{m+2}\sum _{j=1}^m\left(\genfrac{}{}{0ex}{}{m+2}{j}\right){\mathrm{\Delta }}_{m-j+1}{B}_j(\langle x\rangle )\hfill \\ =\sum _{k=0}^m{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\mathit{\text{for }}x\notin \mathbb{Z};\hfill \\ \frac{1}{m+2}{\mathrm{\Delta }}_{m+1}+\frac{1}{m+2}\sum _{j=2}^m\left(\genfrac{}{}{0ex}{}{m+2}{j}\right){\mathrm{\Delta }}_{m-j+1}{B}_j(\langle x\rangle )\hfill \\ =\sum _{k=0}^m{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Delta }}_m,\mathit{\text{for }}x\in \mathbb{Z}.\hfill \end{array}$$

Fourier series of functions of the second type

Let ${\beta }_m(x)={\sum }_{k=0}^m\frac{1}{k!(m-k)!}{b}_k(x){\mathbb{B}}_{m-k}^{(r+1)}(x)$, where $r,m$ are integers with $m\ge 1$. Then we will investigate the function

$${\beta }_m(\langle x\rangle )=\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),$$

defined on $\mathbb{R}$, which is periodic with period 1.

The Fourier series of ${\beta }_m(\langle x\rangle )$ is

$$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{B}_n^{(m)}{e}^{2\pi inx},$$

where

$$\begin{array}{rcl}{B}_n^{(m)}& =& {\int }_0^1{\beta }_m(\langle x\rangle )e^{-2\pi inx}dx\\ & =& {\int }_0^1{\beta }_m(x)e^{-2\pi inx}dx.\end{array}$$

Before proceeding further, we note the following.

$$\begin{array}{rcl}{\beta }_m^{\prime }(x)& =& \sum _{k=0}^m\{\frac{k}{k!(m-k)!}{b}_{k-1}(x){\mathbb{B}}_{m-k}^{(r+1)}(x)+\frac{m-k}{k!(m-k)!}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\}\\ & =& \sum _{k=1}^m\frac{1}{(k-1)!(m-k)!}{b}_{k-1}(x){\mathbb{B}}_{m-k}^{(r+1)}(x)+\sum _{k=0}^{m-1}\frac{1}{k!(m-k-1)!}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& \sum _{k=0}^{m-1}\frac{1}{k!(m-k-1)!}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)+\sum _{k=0}^{m-1}\frac{1}{k!(m-1-k)!}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& 2{\beta }_{m-1}(x).\end{array}$$

Thus

$${\left(\frac{{\beta }_{m+1}(x)}{2}\right)}^{\prime }={\beta }_m(x),\text{and}{\int }_0^1{\beta }_m(x)dx=\frac{1}{2}({\beta }_{m+1}(1)-{\beta }_{m+1}(0)).$$

For $m\ge 1$, we put

$$\begin{array}{rcl}{\mathrm{\Omega }}_m& =& {\beta }_m(1)-{\beta }_m(0)\\ & =& \sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k(1){\mathbb{B}}_{m-k}^{(r+1)}(1)-\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& \sum _{k=0}^{m-1}\frac{1}{k!(m-k)!}(2b_k-{\delta }_{k,0})({\mathbb{B}}_{m-k}^{(r+1)}+{\mathbb{B}}_{m-k-1}^{(r)})\\ & & +\frac{1}{m!}(2b_m-{\delta }_{m,0})-\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& 2\sum _{k=0}^{m-1}\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=0}^{m-1}\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k-1}^{(r)}\\ & & -\frac{1}{m!}{\mathbb{B}}_m^{(r+1)}-\frac{1}{m!}{\mathbb{B}}_{m-1}^{(r)}+\frac{1}{m!}{b}_m-\sum _{k=0}^{m-1}\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}\\ & =& \sum _{k=1}^m\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=0}^{m-1}\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k-1}^{(r)}-\frac{1}{m!}{\mathbb{B}}_{m-1}^{(r)}.\end{array}$$

Hence

$${\beta }_m(0)={\beta }_m(1)⟺{\mathrm{\Omega }}_m=0,\text{and}{\int }_0^1{\beta }_m(x)dx=\frac{1}{2}{\mathrm{\Omega }}_{m+1}.$$

We now would like to determine the Fourier coefficients $B_n^{(m)}$.

Case 1: $n\ne 0$.

$$\begin{array}{rcl}{B}_n^{(m)}& =& {\int }_0^1{\beta }_m(x)e^{-2\pi inx}dx\\ & =& -\frac{1}{2\pi in}{[{\beta }_m(x)e^{-2\pi inx}]}_0^1+\frac{1}{2\pi in}{\int }_0^1{\beta }_m^{\prime }(x)e^{-2\pi inx}dx\\ & =& -\frac{1}{2\pi in}({\beta }_m(1)-{\beta }_m(0))+\frac{2}{2\pi in}{\int }_0^1{\beta }_{m-1}(x)e^{-2\pi inx}dx\\ & =& \frac{2}{2\pi in}{B}_n^{(m-1)}-\frac{1}{2\pi in}{\mathrm{\Omega }}_m\\ & =& \frac{2}{2\pi in}(\frac{2}{2\pi in}{B}_n^{(m-2)}-\frac{1}{2\pi in}{\mathrm{\Omega }}_{m-1})-\frac{1}{2\pi in}{\mathrm{\Omega }}_m\\ & =& {\left(\frac{2}{2\pi in}\right)}^2{B}_n^{(m-2)}-\sum _{j=1}^2\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1}\\ & =& \cdots \\ & =& {\left(\frac{2}{2\pi in}\right)}^m{B}_n^{(0)}-\sum _{j=1}^m\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1}\\ & =& -\sum _{j=1}^m\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1}.\end{array}$$

Case 2: $n=0$.

$$B_0^{(m)}={\int }_0^1{\beta }_m(x)dx=\frac{1}{2}{\mathrm{\Omega }}_{m+1}.$$

${\beta }_m(\langle x\rangle )$, $(m\ge 1)$ is piecewise $C^{\mathrm{\infty }}$. Moreover, ${\beta }_m(\langle x\rangle )$ is continuous for those positive integers m with ${\mathrm{\Omega }}_m=0$ and discontinuous with jump discontinuities at integers for those positive integers m with ${\mathrm{\Omega }}_m\ne 0$.

Assume first that m is a positive integer with ${\mathrm{\Omega }}_m=0$. Then ${\beta }_m(0)={\beta }_m(1)$. Thus ${\beta }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and continuous. Hence the Fourier series of ${\beta }_m(\langle x\rangle )$ converges uniformly to ${\beta }_m(\langle x\rangle )$, and

$$\begin{array}{rcl}{\beta }_m(\langle x\rangle )& =& \frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\sum _{j=1}^m\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1})e^{2\pi inx}\\ & =& \frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{j=1}^m\frac{2^{j-1}}{j!}{\mathrm{\Omega }}_{m-j+1}(-j!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^j})\\ & =& \frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{j=2}^m\frac{2^{j-1}}{j!}{\mathrm{\Omega }}_{m-j+1}{B}_j(\langle x\rangle )\\ & & +{\mathrm{\Omega }}_m\times \{\begin{array}{cc}{B}_1(\langle x\rangle ),\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0,\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}\end{array}$$

Now, we can state our first result.

Theorem 3.1

For each positive integer l, we let

$${\mathrm{\Omega }}_l=\sum _{k=1}^l\frac{1}{k!(l-k)!}{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=0}^{l-1}\frac{1}{k!(l-k)!}{b}_k{\mathbb{B}}_{l-k-1}^{(r)}-\frac{1}{l!}{\mathbb{B}}_{l-1}^{(r)}.$$

Assume that ${\mathrm{\Omega }}_m=0$ for a positive integer m. Then we have the following.

1. ${\sum }_{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )$ has the Fourier series expansion

   $$\begin{array}{c}\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )\hfill \\ =\frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\sum _{j=1}^m\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1})e^{2\pi inx},\hfill \end{array}$$

   for all $x\in \mathbb{R}$, where the convergence is uniform.
2. $$\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )=\frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{j=2}^m\frac{2^{j-1}}{j!}{\mathrm{\Omega }}_{m-j+1}{B}_j(\langle x\rangle ),$$

   for all $x\in \mathbb{R}$, where $B_j(\langle x\rangle )$ is the Bernoulli function.

Assume next that ${\mathrm{\Omega }}_m\ne 0$ for a positive integer m. Then ${\beta }_m(0)\ne {\beta }_m(1)$. So ${\beta }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and discontinuous with jump discontinuities at integers. The Fourier series of ${\beta }_m(\langle x\rangle )$ converges pointwise to ${\beta }_m(\langle x\rangle )$, for $x\notin \mathbb{Z}$, and converges to

$$\frac{1}{2}({\beta }_m(0)+{\beta }_m(1))={\beta }_m(0)+\frac{1}{2}{\mathrm{\Omega }}_m,$$

for $x\in \mathbb{Z}$.

Now, we can state our second theorem.

Theorem 3.2

For each positive integer l, we let

$${\mathrm{\Omega }}_l=\sum _{k=1}^l\frac{1}{k!(l-k)!}{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=0}^{l-1}\frac{1}{k!(l-k)!}{b}_k{\mathbb{B}}_{l-k-1}^{(r)}-\frac{1}{l!}{\mathbb{B}}_{l-1}^{(r)}.$$

Assume that ${\mathrm{\Omega }}_m\ne 0$ for a positive integer m. Then we have the following.

1. $$\begin{array}{c}\frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}(-\sum _{j=1}^m\frac{2^{j-1}}{{(2\pi in)}^j}{\mathrm{\Omega }}_{m-j+1})e^{2\pi inx}\hfill \\ =\{\begin{array}{cc}{\sum }_{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\hfill & \mathit{\text{for }}x\notin \mathbb{Z},\hfill \\ {\sum }_{k=0}^m\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Omega }}_m,\hfill & \mathit{\text{for }}x\in \mathbb{Z}.\hfill \end{array}\hfill \end{array}$$
2. $$\begin{array}{c}\frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{j=1}^m\frac{2^{j-1}}{j!}{\mathrm{\Omega }}_{m-j+1}{B}_j(\langle x\rangle )\hfill \\ =\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\mathit{\text{for }}x\notin \mathbb{Z};\hfill \\ \frac{1}{2}{\mathrm{\Omega }}_{m+1}+\sum _{j=2}^m\frac{2^{j-1}}{j!}{\mathrm{\Omega }}_{m-j+1}{B}_j(\langle x\rangle )\hfill \\ =\sum _{k=0}^m\frac{1}{k!(m-k)!}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Omega }}_m,\mathit{\text{for }}x\in \mathbb{Z}.\hfill \end{array}$$

Fourier series of functions of the third type

Let

$${\gamma }_m(x)=\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(x){\mathbb{B}}_{m-k}^{(r+1)}(x),$$

where $r,m$ are integers with $m\ge 2$. Then we will consider the function

$${\gamma }_m(\langle x\rangle )=\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),$$

defined on $\mathbb{R}$, which is periodic with period 1.

The Fourier series of ${\gamma }_m(\langle x\rangle )$ is

$$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_n^{(m)}{e}^{2\pi inx},$$

where

$$\begin{array}{rcl}{C}_n^{(m)}& =& {\int }_0^1{\gamma }_m(\langle x\rangle )e^{-2\pi inx}dx\\ & =& {\int }_0^1{\gamma }_m(x)e^{-2\pi inx}dx.\end{array}$$

Before proceeding further, we need to observe the following.

$$\begin{array}{rcl}{\gamma }_m^{\prime }(x)& =& \sum _{k=1}^{m-1}\frac{1}{m-k}{b}_{k-1}(x){\mathbb{B}}_{m-k}^{(r+1)}(x)+\sum _{k=1}^{m-1}\frac{1}{k}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& \sum _{k=0}^{m-2}\frac{1}{m-k-1}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)+\sum _{k=1}^{m-1}\frac{1}{k}{b}_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)\\ & =& \sum _{k=1}^{m-2}(\frac{1}{m-k-1}+\frac{1}{k})b_k(x){\mathbb{B}}_{m-k-1}^{(r+1)}(x)+\frac{1}{m-1}{\mathbb{B}}_{m-1}^{(r+1)}(x)+\frac{1}{m-1}{b}_{m-1}(x)\\ & =& (m-1){\gamma }_{m-1}(x)+\frac{1}{m-1}{\mathbb{B}}_{m-1}^{(r+1)}(x)+\frac{1}{m-1}{b}_{m-1}(x).\end{array}$$

From this, we see that

$${\left(\frac{1}{m}({\gamma }_{m+1}(x)-\frac{1}{m(m+1)}{\mathbb{B}}_{m+1}^{(r+1)}(x)-\frac{1}{m(m+1)}{b}_{m+1}(x))\right)}^{\prime }={\gamma }_m(x),$$

and

$$\begin{array}{c}{\int }_0^1{\gamma }_m(x)dx\hfill \\ =\frac{1}{m}{[{\gamma }_{m+1}(x)-\frac{1}{m(m+1)}{\mathbb{B}}_{m+1}^{(r+1)}(x)-\frac{1}{m(m+1)}{b}_{m+1}(x)]}_0^1\hfill \\ =\frac{1}{m}({\gamma }_{m+1}(1)-{\gamma }_m(0)-\frac{1}{m(m+1)}({\mathbb{B}}_{m+1}^{(r+1)}(1)-{\mathbb{B}}_{m+1}^{(r+1)}(0))\hfill \\ -\frac{1}{m(m+1)}(b_{m+1}(1)-b_{m+1}(0)))\hfill \\ =\frac{1}{m}({\gamma }_{m+1}(1)-{\gamma }_{m+1}(0)-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1}).\hfill \end{array}$$

For $m\ge 2$, we let

$$\begin{array}{rcl}{\mathrm{\Lambda }}_m& =& {\gamma }_m(1)-{\gamma }_m(0)\\ & =& \sum _{k=1}^{m-1}\frac{1}{k(m-k)}(b_k(1){\mathbb{B}}_{m-k}^{(r+1)}(1)-b_k{\mathbb{B}}_{m-k}^{(r+1)})\\ & =& \sum _{k=1}^{m-1}\frac{1}{k(m-k)}((2b_k-{\delta }_{k,0})({\mathbb{B}}_{m-k}^{(r+1)}+{\mathbb{B}}_{m-k-1}^{(r)})-b_k{\mathbb{B}}_{m-k}^{(r+1)})\\ & =& \sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k-1}^{(r)}.\end{array}$$

Then

$${\gamma }_m(0)={\gamma }_m(1)⟺{\mathrm{\Lambda }}_m=0,$$

and

$${\int }_0^1{\gamma }_m(x)dx=\frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1}).$$

Now, we would like to determine the Fourier coefficients $C_n^{(m)}$.

Case 1: $n\ne 0$. For this computation, we need to know the following:

$$\begin{array}{c}{\int }_0^1{\mathbb{B}}_l^{(r+1)}(x)e^{-2\pi inx}dx=\{\begin{array}{cc}-{\sum }_{k=1}^l\frac{{(l)}_{k-1}}{{(2\pi in)}^k}{\mathbb{B}}_{l-k}^{(r)},\hfill & \text{for }n\ne 0,\hfill \\ \frac{1}{l+1}{\mathbb{B}}_l^{(r)},\hfill & \text{for }n=0,\hfill \end{array}\hfill \\ {\int }_0^1{b}_l(x)e^{-2\pi inx}dx=\{\begin{array}{cc}-{\sum }_{k=1}^l\frac{{(l)}_{k-1}}{{(2\pi in)}^k}{b}_{l-k+1},\hfill & \text{for }n\ne 0,\hfill \\ \frac{1}{l+1}{b}_{l+1},\hfill & \text{for }n=0,\hfill \end{array}\hfill \\ C_n^{(m)}={\int }_0^1{\gamma }_m(x)e^{-2\pi inx}dx\hfill \\ \phantom{C_n^{(m)}}=-\frac{1}{2\pi in}{[{\gamma }_m(x)e^{-2\pi inx}]}_0^1+\frac{1}{2\pi in}{\int }_0^1{\gamma }_m^{\prime }(x)e^{-2\pi inx}dx\hfill \\ \phantom{C_n^{(m)}}=-\frac{1}{2\pi in}({\gamma }_m(1)-{\gamma }_m(0))\hfill \\ \phantom{C_n^{(m)}=}+\frac{1}{2\pi in}{\int }_0^1((m-1){\gamma }_{m-1}(x)+\frac{1}{m-1}{\mathbb{B}}_{m-1}^{(r+1)}(x)+\frac{1}{m-1}{b}_{m-1}(x))e^{-2\pi inx}dx\hfill \\ \phantom{C_n^{(m)}}=\frac{m-1}{2\pi in}{C}_n^{(m-1)}-\frac{1}{2\pi in}{\mathrm{\Lambda }}_m+\frac{1}{2\pi in(m-1)}{\int }_0^1{\mathbb{B}}_{m-1}^{(r+1)}(x)e^{-2\pi inx}dx\hfill \\ \phantom{C_n^{(m)}=}+\frac{1}{2\pi in(m-1)}{\int }_0^1{b}_{m-1}(x)e^{-2\pi inx}dx\hfill \\ \phantom{C_n^{(m)}}=\frac{m-1}{2\pi in}{C}_n^{(m-1)}-\frac{1}{2\pi in}{\mathrm{\Lambda }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Theta }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Phi }}_m,\hfill \end{array}$$

where

$$\begin{array}{c}{\mathrm{\Lambda }}_m={\gamma }_m(1)-{\gamma }_m(0)\hfill \\ \phantom{{\mathrm{\Lambda }}_m}=\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+2\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k-1}^{(r)},\hfill \\ {\mathrm{\Theta }}_m=\sum _{k=1}^{m-1}\frac{{(m-1)}_{k-1}}{{(2\pi in)}^k}{\mathbb{B}}_{m-k-1}^{(r)},{\mathrm{\Phi }}_m=\sum _{k=1}^{m-1}\frac{{(m-1)}_{k-1}}{{(2\pi in)}^k}{b}_{m-k}.\hfill \\ C_n^{(m)}=\frac{m-1}{2\pi in}{C}_n^{(m-1)}-\frac{1}{2\pi in}{\mathrm{\Lambda }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Theta }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Phi }}_m\hfill \\ \phantom{C_n^{(m)}}=\frac{m-1}{2\pi in}(\frac{m-2}{2\pi in}{C}_n^{(m-2)}-\frac{1}{2\pi in}{\mathrm{\Lambda }}_{m-1}-\frac{1}{2\pi in(m-2)}{\mathrm{\Theta }}_{m-1}-\frac{1}{2\pi in(m-2)}{\mathrm{\Phi }}_{m-1})\hfill \\ \phantom{C_n^{(m)}=}-\frac{1}{2\pi in}{\mathrm{\Lambda }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Theta }}_m-\frac{1}{2\pi in(m-1)}{\mathrm{\Phi }}_m\hfill \\ \phantom{C_n^{(m)}}=\frac{{(m-1)}_2}{{(2\pi in)}^2}{C}_n^{(m-2)}-\sum _{j=1}^2\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j}{\mathrm{\Lambda }}_{m-j+1}\hfill \\ \phantom{C_n^{(m)}=}-\sum _{j=1}^2\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}{\mathrm{\Theta }}_{m-j+1}-\sum _{j=1}^2\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}{\mathrm{\Phi }}_{m-j+1}\hfill \\ \phantom{C_n^{(m)}}=\cdots \hfill \\ \phantom{C_n^{(m)}}=-\sum _{j=1}^{m-1}\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j}{\mathrm{\Lambda }}_{m-j+1}-\sum _{j=1}^{m-1}\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}{\mathrm{\Theta }}_{m-j+1}-\sum _{j=1}^{m-1}\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}{\mathrm{\Phi }}_{m-j+1}.\hfill \end{array}$$

We note here that

$$\begin{array}{c}\sum _{j=1}^{m-1}\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}{\mathrm{\Phi }}_{m-j+1}\hfill \\ =\sum _{j=1}^{m-1}\frac{{(m-1)}_{j-1}}{{(2\pi in)}^j(m-j)}\sum _{k=1}^{m-j}\frac{{(m-j)}_{k-1}}{{(2\pi in)}^k}{b}_{m-j-k+1}\hfill \\ =\sum _{j=1}^{m-1}\sum _{k=1}^{m-j}\frac{{(m-1)}_{j+k-2}}{{(2\pi in)}^{j+k}(m-j)}{b}_{m-j-k+1}\hfill \\ =\sum _{j=1}^{m-1}\frac{1}{m-j}\sum _{k=1}^{m-j}\frac{{(m-1)}_{j+k-2}}{{(2\pi in)}^{j+k}}{b}_{m-j-k+1}\hfill \\ =\sum _{j=1}^{m-1}\frac{1}{m-j}\sum _{s=j+1}^m\frac{{(m-1)}_{s-2}}{{(2\pi in)}^s}{b}_{m-s+1}\hfill \\ =\sum _{s=2}^m\frac{{(m-1)}_{s-2}}{{(2\pi in)}^s}{b}_{m-s+1}\sum _{j=1}^{s-1}\frac{1}{m-j}\hfill \\ =\sum _{s=1}^m\frac{{(m-1)}_{s-2}}{{(2\pi in)}^s}{b}_{m-s+1}(H_{m-1}-H_{m-s})\hfill \\ =\frac{1}{m}\sum _{s=1}^m\frac{{(m)}_s}{{(2\pi in)}^s}\frac{H_{m-1}-H_{m-s}}{m-s+1}{b}_{m-s+1}.\hfill \end{array}$$

Putting everything together, we get

$$C_n^{(m)}=-\frac{1}{m}\sum _{s=1}^m\frac{{(m)}_s}{{(2\pi in)}^s}\{{\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1})\}.$$

Case 2: $n=0$.

$$\begin{array}{rcl}{C}_0^{(m)}& =& {\int }_0^1{\gamma }_m(x)dx\\ & =& \frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1}).\end{array}$$

${\gamma }_m(\langle x\rangle )$, $m\ge 2$ is piecewise $C^{\mathrm{\infty }}$. Moreover, ${\gamma }_m(\langle x\rangle )$ is continuous for those integers $m\ge 2$ with ${\mathrm{\Lambda }}_m=0$ and discontinuous with jump discontinuities at integers for those integers $m\ge 2$ with ${\mathrm{\Lambda }}_m\ne 0$.

Assume first that ${\mathrm{\Lambda }}_m=0$. Then ${\gamma }_m(0)={\gamma }_m(1)$. Thus ${\gamma }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and continuous. Hence the Fourier series of ${\gamma }_m(\langle x\rangle )$ converges uniformly to ${\gamma }_m(\langle x\rangle )$, and

$$\begin{array}{rcl}{\gamma }_m(\langle x\rangle )& =& \frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1})\\ & & +\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\{-\frac{1}{m}\sum _{s=1}^m\frac{{(m)}_s}{{(2\pi in)}^s}({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))\}{e}^{2\pi inx}\\ & =& \frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1})\\ & & +\frac{1}{m}\sum _{s=1}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))(-s!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^s})\\ & =& \frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1})\\ & & +\frac{1}{m}\sum _{s=2}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(\langle x\rangle )\\ & & +{\mathrm{\Lambda }}_m\times \{\begin{array}{cc}{B}_1(\langle x\rangle ),\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0,\hfill & \text{for }x\in \mathbb{Z}\hfill \end{array}\\ & =& \frac{1}{m}\sum _{\begin{array}{c}s=0\\ s\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(\langle x\rangle )\\ & & +{\mathrm{\Lambda }}_m\times \{\begin{array}{cc}{B}_1(\langle x\rangle ),\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0,\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}\end{array}$$

Now, we can state our first result.

Theorem 4.1

For each integer $l\ge 2$, we let

$${\mathrm{\Lambda }}_l=\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k-1}^{(r)},$$

with ${\mathrm{\Lambda }}_1=0$. Assume that ${\mathrm{\Lambda }}_m=0$ for an integer $m\ge 2$. Then we have the following.

1. ${\sum }_{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )$ has Fourier series expansion

   $$\begin{array}{c}\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )\hfill \\ =\frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1})\hfill \\ +\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\{-\frac{1}{m}\sum _{s=1}^m\frac{{(m)}_s}{{(2\pi in)}^s}({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))\}{e}^{2\pi inx},\hfill \end{array}$$

   for all $x\in \mathbb{R}$, where the convergence is uniform.
2. $$\begin{array}{c}\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle )\hfill \\ =\frac{1}{m}\sum _{\begin{array}{c}s=0\\ s\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(\langle x\rangle ),\hfill \end{array}$$

   for all $x\in \mathbb{R}$, where $B_s(\langle x\rangle )$ is the Bernoulli function.

Assume next that m is an integer ≥2 with ${\mathrm{\Lambda }}_m\ne 0$. Then ${\gamma }_m(0)\ne {\gamma }_m(1)$. Hence ${\gamma }_m(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and discontinuous with jump discontinuities at integers. Then the Fourier series of ${\gamma }_m(\langle x\rangle )$ converges pointwise to ${\gamma }_m(\langle x\rangle )$, for $x\notin \mathbb{Z}$, and converges to

$$\begin{array}{rcl}\frac{1}{2}({\gamma }_m(0)+{\gamma }_m(1))& =& {\gamma }_m(0)+\frac{1}{2}{\mathrm{\Lambda }}_m\\ & =& \sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Lambda }}_m,\end{array}$$

for $x\in \mathbb{Z}$.

Now, we can state our second result.

Theorem 4.2

For each integer $l\ge 2$, let

$${\mathrm{\Lambda }}_l=\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k}^{(r+1)}+2\sum _{k=1}^{l-1}\frac{1}{k(l-k)}{b}_k{\mathbb{B}}_{l-k-1}^{(r)},$$

with ${\mathrm{\Lambda }}_1=0$. Assume that ${\mathrm{\Lambda }}_m\ne 0$ for an integer $m\ge 2$. Then we have the following.

1. $$\begin{array}{c}=\frac{1}{m}({\mathrm{\Lambda }}_{m+1}-\frac{1}{m(m+1)}{\mathbb{B}}_m^{(r)}-\frac{1}{m(m+1)}{b}_{m+1})\hfill \\ +\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\{-\frac{1}{m}\sum _{s=1}^m\frac{{(m)}_s}{{(2\pi in)}^s}({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))\}{e}^{2\pi inx}\hfill \\ =\{\begin{array}{cc}{\sum }_{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\hfill & \mathit{\text{for }}x\notin \mathbb{Z},\hfill \\ {\sum }_{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Lambda }}_m,\hfill & \mathit{\text{for }}x\in \mathbb{Z}.\hfill \end{array}\hfill \end{array}$$
2. $$\begin{array}{c}\frac{1}{m}\sum _{s=0}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(\langle x\rangle )\hfill \\ =\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k(\langle x\rangle ){\mathbb{B}}_{m-k}^{(r+1)}(\langle x\rangle ),\mathit{\text{for }}x\notin \mathbb{Z};\hfill \\ \frac{1}{m}\sum _{\begin{array}{c}s=0\\ s\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{s}\right)({\mathrm{\Lambda }}_{m-s+1}+\frac{H_{m-1}-H_{m-s}}{m-s+1}({\mathbb{B}}_{m-s}^{(r)}+b_{m-s+1}))B_s(\langle x\rangle )\hfill \\ =\sum _{k=1}^{m-1}\frac{1}{k(m-k)}{b}_k{\mathbb{B}}_{m-k}^{(r+1)}+\frac{1}{2}{\mathrm{\Lambda }}_m,\mathit{\text{for }}x\in \mathbb{Z}.\hfill \end{array}$$

Results and discussion

In this paper, we study three types of sums of products of ordered Bell and poly-Bernoulli functions and derive their Fourier series expansion. In addition, we express those functions in terms of Bernoulli functions. The Fourier series expansion of the ordered Bell and poly-Bernoulli functions are useful in computing the special values of the poly-zeta and multiple zeta function. For details, one is referred to [3, 7–18]. It is expected that the Fourier series of the ordered Bell functions will find some applications in connection with a certain generalization of the Euler zeta function and the higher-order generalized Frobenius-Euler numbers and polynomials.

Conclusion

In this paper, we considered the Fourier series expansion of the ordered Bell and poly-Bernoulli functions which are obtained by extending by periodicity of period 1 the ordered Bell and poly-Bernoulli polynomials on $[0,1)$. The Fourier series are explicitly determined.