Fourier series of higher-order Daehee and Changhee functions and their applications

Abstract

In the paper, the author considers the Fourier series related to higher-order Daehee and Changhee functions and establishes some new identities for higher-order Daehee and Changhee functions.

Introduction and main results

It is common knowledge that the Bernoulli polynomials $B_n(x)$ and the Euler polynomials $E_n(x)$ for $n\ge 0$ can be generated by

$$\begin{array}{rl}\frac{t}{e^t-1}{e}^{xt}& =\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}\end{array}$$

and

$$\begin{array}{r}\frac{2}{e^t+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!},\end{array}$$

respectively (see [1–23]).

With the viewpoint of deformed Bernoulli polynomials, the Daehee polynomials $D_n(x)$ for $n\ge 0$ are defined by the generating function to be

$$\frac{log(1+t)}{t}{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{D}_n(x)\frac{t^n}{n!}.$$ 1

It is easy to see that the generating function of the Daehee polynomials $D_n(x)$ can be reformed as

$$\frac{log(1+t)}{t}{(1+t)}^x=\frac{log(1+t)}{e^{log(1+t)}-1}{e}^{xlog(1+t)}.$$

From (1), we note that

$$\begin{array}{rl}\frac{log(1+t)}{e^{log(1+t)}-1}{e}^{xlog(1+t)}& =\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{1}{n!}{(log(1+t))}^n\\ & =\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\sum _{m=n}^{\mathrm{\infty }}{S}_1(m,n)\frac{t^m}{m!}\\ & =\sum _{m=0}^{\mathrm{\infty }}(\sum _{n=0}^m{B}_n(x)S_1(m,n))\frac{t^m}{m!},\end{array}$$ 2

where $S_1(m,n)$ stands for the Stirling number of the first kind which is defined as

$${(x)}_0=1,{(x)}_n=x(x-1)\cdots (x-n+1)=\sum _{l=0}^n{S}_1(n,l)x^l(n\ge 1).$$

Combining (1) with (2) yields the following relation:

$$D_m(x)=\sum _{n=0}^m{B}_n(x)S_1(m,n)(m\ge 0).$$

By replacing t by $e^t-1$ in (1), we can derive

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}& =\frac{t}{e^t-1}{e}^{xt}=\sum _{m=0}^{\mathrm{\infty }}{D}_m(x)\frac{1}{m!}{(e^t-1)}^m\\ & =\sum _{m=0}^{\mathrm{\infty }}{D}_m(x)\sum _{n=m}^{\mathrm{\infty }}{S}_2(n,m)\frac{t^n}{n!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{m=0}^n{D}_n(x)S_2(n,m))\frac{t^n}{n!},\end{array}$$ 3

where $S_2(n,m)$ is the Stirling number of the second kind which is given by $x^n={\sum }_{l=0}^{\mathrm{\infty }}{S}_2(n,l){(x)}_l$ $(n\ge 0)$.

Comparing the coefficients on the both sides of (3), we obtain

$$B_n(x)=\sum _{m=0}^n{D}_m(x)S_2(n,m)(n\ge 0).$$

Also, with the viewpoint of deformed Euler polynomials, the Changhee polynomials ${\mathit{Ch}}_n(x)$ for $n\ge 0$ are defined by the generating function to be

$$\frac{2}{2+t}{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{\mathit{Ch}}_n(x)\frac{t^n}{n!}.$$ 4

Definition (4) can be written as

$$\begin{array}{rl}\frac{2}{e^{log(1+t)}+1}{e}^{xlog(1+t)}& =\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{1}{n!}{(log(1+t))}^n\\ & =\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\sum _{m=n}^{\mathrm{\infty }}{S}_1(m,n)\frac{t^m}{m!}\\ & =\sum _{m=0}^{\mathrm{\infty }}(\sum _{n=0}^m{E}_n(x)S_1(m,n))\frac{t^m}{m!}.\end{array}$$

Combination of this identity with (4) results in the following relation:

$${\mathit{Ch}}_m(x)=\sum _{n=0}^m{E}_n(x)S_1(m,n)(m\ge 0).$$

Now replacing t by $e^t-1$ in (4), we have

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!}& =\frac{2}{e^t+1}{e}^{xt}=\sum _{m=0}^{\mathrm{\infty }}{\mathit{Ch}}_m(x)\frac{1}{m!}{(e^t-1)}^m\\ & =\sum _{m=0}^{\mathrm{\infty }}{\mathit{Ch}}_m(x)\sum _{n=m}^{\mathrm{\infty }}{S}_2(n,m)\frac{t^n}{n!}\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{m=0}^n{\mathit{Ch}}_n(x)S_2(n,m))\frac{t^n}{n!}.\end{array}$$

Equating coefficients on the very ends of the above identity leads to

$$E_n(x)=\sum _{m=0}^n{\mathit{Ch}}_m(x)S_2(n,m)(n\ge 0).$$

In recent decades, many mathematicians have investigated some interesting extensions or modifications of the Daehee and Changhee polynomials along with related combinatorial identities and their applications (see [4, 9, 10, 14, 16, 17, 19, 23]). Especially, Kim and his coauthors have studied the Fourier series related to various types of Bernoulli functions in [7, 11–13, 15]. The purpose of this paper is to study the Fourier series related to higher-order Daehee and Changhee functions and establish some new identities for higher-order Daehee and Changhee functions.

For any real number x, we define

$$\langle x\rangle =x-[x]\in (0,1),$$

where $[x]$ is the integer part of x. Then $D_n(\langle x\rangle )$ are functions defined on $(-\mathrm{\infty },\mathrm{\infty })$ and periodic with period 1, which are called Daehee functions.

For $r\in \mathbb{N}$ and $n\ge 0$, we note that the higher-order Daehee polynomials $D_n^{(r)}(x)$ and the higher-order Changhee polynomials ${\mathit{Ch}}_n^{(r)}(x)$ may also be represented by the following generating function:

$${\left(\frac{log(1+t)}{t}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{D}_n^{(r)}(x)\frac{t^n}{n!}$$ 5

and

$${\left(\frac{2}{2+t}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{\mathit{Ch}}_n^{(r)}(x)\frac{t^n}{n!},$$ 6

respectively (see [4, 10, 14]). When $x=0$, $D_n^{(r)}=D_n^{(r)}(0)$ are called the higher-order Daehee numbers and ${\mathit{Ch}}_n^{(r)}={\mathit{Ch}}_n^{(r)}(0)$ are called the higher-order Changhee numbers. And it is easy to see that

$$D_n^{(1)}(x)=D_n(x),{\mathit{Ch}}_n^{(1)}(x)={\mathit{Ch}}_n(x).$$

Then $D_n^{(r)}(\langle x\rangle )$ and ${\mathit{Ch}}_n^{(r)}(\langle x\rangle )$ are functions defined on $(-\mathrm{\infty },\mathrm{\infty })$ and periodic of period 1, which are called Daehee functions of order r and Changhee functions of order r, respectively.

Recall from [15, 24] that the Bernoulli function may be represented by

$$\begin{array}{r}{B}_m(\langle x\rangle )=-m!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^m}(m\ge 2)\end{array}$$ 7

and

$$\begin{array}{r}-m!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^m}=\{\begin{array}{cc}{B}_1(\langle x\rangle )\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}\end{array}$$ 8

The Fourier series expansion of the Bernoulli functions is useful in computing the special values of the Dirichlet L-functions. For details, one is referred to [24].

Our main results in this paper can be stated as the following theorems.

Theorem 1

Let $m\ge 2$, $r\ge 1$. Assume that $D_{m-1}^{(r)}=0$.

1. $D_m^{(r)}(\langle x\rangle )$ has the Fourier series expansion

   $$\begin{array}{r}{D}_m^{(r)}(\langle x\rangle )=D_m^{(r)}-\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\left(\sum _{k=1}^m\frac{{(m)}_k}{{(2\pi in)}^k}{D}_{m-k}^{(r)}\right)e^{2\pi inx}\end{array}$$

   for $x\in (-\mathrm{\infty },\mathrm{\infty })$. Here the convergence is uniform.

2. $D_m^{(r)}(\langle x\rangle )={\sum }_{\begin{array}{c}k=0\\ k\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-1}^{(r)}{B}_k(\langle x\rangle )$, for all $x\in (-\mathrm{\infty },\mathrm{\infty })$, where $B_k(\langle x\rangle )$ is the Bernoulli function.

Theorem 2

Let $m\ge 2$, $r\ge 1$. Assume that $D_{m-1}^{(r)}\ne 0$.

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

   Here the convergence is pointwise.
2. $$\begin{array}{r}\sum _{k=0}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-k}^{(r)}{B}_k(\langle x\rangle )=D_m^{(r)}(x)\mathit{\text{for }}x\notin \mathbb{Z}\end{array}$$

   and

   $$\begin{array}{r}\sum _{\begin{array}{c}k=0\\ k\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-k}^{(r)}{B}_k(\langle x\rangle )=D_m^{(r)}+\frac{m}{2}{D}_{m-1}^{(r)}\mathit{\text{for }}x\in \mathbb{Z},\end{array}$$

   where $B_k(\langle x\rangle )$ is the Bernoulli function.

Theorem 3

Let $m\ge 2$, $r\ge 1$. Assume that ${\mathit{Ch}}_m^{(r)}={\mathit{Ch}}_m^{(r-1)}$.

1. ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ has the Fourier series expansion

   $$\begin{array}{rl}{\mathit{Ch}}_m^{(r)}(\langle x\rangle )=& \frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})\\ & +\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\left(\sum _{k=1}^m\frac{2{(m)}_{k-1}}{{(2\pi in)}^k}({\mathit{Ch}}_{m-k+1}^{(r)}-{\mathit{Ch}}_{m-k+1}^{(r-1)})\right)e^{2\pi inx}\end{array}$$

   for $x\in (-\mathrm{\infty },\mathrm{\infty })$. Here the convergence is uniform.
2. $$\begin{array}{rl}{\mathit{Ch}}_m^{(r)}(\langle x\rangle )=& \frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r)}-{\mathit{Ch}}_{m+1}^{(r)})\\ & +\sum _{k=1}^m\frac{2{(m)}_{k-1}}{k!}({\mathit{Ch}}_{m-k+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})B_k(\langle x\rangle )\mathit{\text{for }}x\notin \mathbb{Z}\end{array}$$

   and

   $$\begin{array}{rl}{\mathit{Ch}}_m^{(r)}(\langle x\rangle )=& \frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m-k+1}^{(r)})\\ & +\sum _{k=2}^m\frac{2{(m)}_{k-1}}{k!}({\mathit{Ch}}_{m-k+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})B_k(\langle x\rangle )\mathit{\text{for }}x\in \mathbb{Z},\end{array}$$

   where $B_k(\langle x\rangle )$ is the Bernoulli function.

Theorem 4

Let $m\ge 1$, $r\ge 1$. Assume that ${\mathit{Ch}}_m^{(r)}\ne {\mathit{Ch}}_m^{(r-1)}$.

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

   Here the convergence is pointwise.
2. $$\begin{array}{c}\frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})+{\sum }_{k=1}^m\frac{2{(m)}_{k-1}}{k!}({\mathit{Ch}}_{m-k+1}^{(r-1)}-{\mathit{Ch}}_{m-k+1}^{(r)})B_k(\langle x\rangle )\\ ={\mathit{Ch}}_m^{(r)}(\langle x\rangle )\mathit{\text{for }}x\notin \mathbb{Z}\end{array}$$

   and

   $$\begin{array}{c}\frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})+{\sum }_{k=2}^m\frac{2{(m)}_{k-1}}{k!}({\mathit{Ch}}_{m-k+1}^{(r-1)}-{\mathit{Ch}}_{m-k+1}^{(r)})B_k(\langle x\rangle )\\ ={\mathit{Ch}}_m^{(r-1)}(\langle x\rangle )\mathit{\text{for }}x\in \mathbb{Z},\end{array}$$

   where $B_k(\langle x\rangle )$ is the Bernoulli function.

Proofs of Theorems 1-4

We are now in a position to prove our four theorems.

By analyzing definition (5), we have

$$D_m^{(r)}(x+1)=D_m^{(r)}(x)+m{D}_{m-1}^{(r)}(x)(m\ge 0).$$

Furthermore, we observe that

$$\begin{array}{rl}\sum _{m=0}^{\mathrm{\infty }}{D}_m^{(r)}(x)\frac{t^m}{m!}& ={\left(\frac{log(1+t)}{t}\right)}^r{(1+t)}^{x+1}\\ & ={\left(\frac{log(1+t)}{t}\right)}^r{(1+t)}^x+{\left(\frac{log(1+t)}{t}\right)}^r{(1+t)}^{x}t\\ & =\sum _{m=0}^{\mathrm{\infty }}{D}_m^{(r)}(x)\frac{t^m}{m!}+\sum _{m=0}^{\mathrm{\infty }}{D}_m^{(r)}(x)\frac{t^{m+1}}{m!}\\ & =\sum _{m=0}^{\mathrm{\infty }}(D_m^{(r)}(x)+m{D}_{m-1}^{(r)}(x))\frac{t^m}{m!}.\end{array}$$

Letting $x=0$ in the above equation leads to

$$D_m^{(r)}(1)=D_m^{(r)}+m{D}_{m-1}^{(r)}(m\ge 0).$$

Now, we assume that $m,r\ge 1$. $D_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$. Further, in view of (2), $D_m^{(r)}(\langle x\rangle )$ is continuous for those $(r,m)$ with $D_{m-1}^{(r)}=0$, and is discontinuous with jump discontinuities at integers for those $(r,m)$ with $D_{m-1}^{(r)}\ne 0$. The Fourier series of $D_m^{(r)}(\langle x\rangle )$ may be represented by

$$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_n^{(r,m)}{e}^{2\pi inx}(i=\sqrt{-1}),$$

where

$$\begin{array}{rl}{C}_n^{(r,m)}& ={\int }_0^1{D}_m^{(r)}(\langle x\rangle )e^{-2\pi inx}dx={\int }_0^1{D}_m^{(r)}(x)e^{-2\pi inx}dx\\ & ={[\frac{1}{m+1}{D}_{m+1}^{(r)}(x)e^{-2\pi inx}]}_0^1+\frac{2\pi in}{m+1}{\int }_0^1{D}_{m+1}^{(r)}(x)e^{-2\pi inx}dx\\ & =\frac{1}{m+1}(D_{m+1}^{(r)}(1)-D_{m+1}^{(r)})+\frac{2\pi in}{m+1}{C}_n^{(r,m+1)}\\ & =D_m^{(r)}+\frac{2\pi in}{m+1}{C}_n^{(r,m+1)}.\end{array}$$ 9

Replacing m by $m-1$ in (9), we arrive at the following result:

$$\begin{array}{r}{C}_n^{(r,m-1)}=D_{m-1}^{(r)}+\frac{2\pi in}{m}{C}_n^{(r,m)}.\end{array}$$

Case 1

Let $n\ne 0$. Then we acquire that

$$\begin{array}{rl}{C}_n^{(r,m)}=& \frac{m}{2\pi in}{C}_n^{(r,m-1)}-\frac{m}{2\pi in}{D}_{m-1}^{(r)}\\ =& \frac{m}{2\pi in}(\frac{m-1}{2\pi in}{C}_n^{(r,m-2)}-\frac{m-1}{2\pi in}{D}_{m-2}^{(r)})-\frac{m}{2\pi in}{D}_{m-1}^{(r)}\\ =& \frac{m(m-1)}{{(2\pi in)}^2}{C}_n^{(r,m-2)}-\frac{m(m-1)}{{(2\pi in)}^2}{D}_{m-2}^{(r)}-\frac{m}{2\pi in}{D}_{m-1}^{(r)}\\ =& \frac{m(m-1)}{{(2\pi in)}^2}(\frac{m-2}{2\pi in}{C}_n^{(r,m-3)}-\frac{m-2}{2\pi in}{D}_{m-3}^{(r)})\\ & -\frac{m(m-1)}{{(2\pi in)}^2}{D}_{m-2}^{(r)}-\frac{m}{2\pi in}{D}_{m-1}^{(r)}\\ =& \frac{m(m-1)(m-2)}{{(2\pi in)}^2}{C}_n^{(r,m-3)}-\frac{m(m-1)(m-2)}{{(2\pi in)}^3}{D}_{m-3}^{(r)}\\ & -\frac{m(m-1)}{{(2\pi in)}^2}{D}_{m-2}^{(r)}-\frac{m}{2\pi in}{D}_{m-1}^{(r)}\\ =& \cdots \\ =& \frac{m(m-1)(m-2)\cdots 2}{{(2\pi in)}^{m-1}}{C}_n^{(r,1)}-\sum _{k=1}^{m-1}\frac{{(m)}_k}{{(2\pi in)}^k}{D}_{m-k}^{(r)}.\end{array}$$ 10

Moreover, we observe that

$$\begin{array}{rl}{C}_n^{(r,1)}& ={\int }_0^1{D}_1^{(r)}(x)e^{-2\pi inx}dx={\int }_0^1(x+D_1^{(r)})e^{-2\pi inx}dx\\ & ={\int }_0^{1}x{e}^{-2\pi inx}dx+D_1^{(r)}{\int }_0^1{e}^{-2\pi inx}dx\\ & =-\frac{1}{2\pi in}{\left[x{e}^{-2\pi inx}\right]}_0^1+\frac{1}{2\pi in}{\int }_0^1{e}^{-2\pi inx}dx=-\frac{1}{2\pi in}.\end{array}$$ 11

Combining (11) with (10), we immediately derive the following equation:

$$C_n^{(r,m)}=\frac{m!}{{(2\pi in)}^m}-\sum _{k=1}^{m-1}\frac{{(m)}_k}{{(2\pi in)}^k}{D}_{m-k}^{(r)}=-\sum _{k=1}^m\frac{{(m)}_k}{{(2\pi in)}^k}{D}_{m-k}^{(r)}.$$

Case 2

Let $n=0$. Then we have

$$\begin{array}{rl}{C}_0^{(r,m)}& ={\int }_0^1{D}_m^{(r)}(\langle x\rangle )dx={\int }_0^1{D}_m^{(r)}(x)dx\\ & =\frac{1}{m+1}{[D_{m+1}^{(r)}(x)]}_0^1\\ & =\frac{1}{m+1}(D_{m+1}^{(r)}(1)-D_{m+1}^{(r)})=D_m^{(r)}.\end{array}$$

While that in (8) converges pointwise, the series in (7) converges uniformly. We assume that $D_{m-1}^{(r)}=0$. Then we have $D_m^{(r)}(1)=D_m^{(r)}$ for $m\ge 2$. As $D_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and continuous, the Fourier series of $D_m^{(r)}(\langle x\rangle )$ converges uniformly to $D_m^{(r)}(\langle x\rangle )$ and

$$\begin{array}{rl}{D}_m^{(r)}(\langle x\rangle )=& \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_n^{(r,m)}{e}^{2\pi inx}\\ =& D_m^{(r)}-\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\left(\sum _{k=1}^m\frac{{(m)}_k}{{(2\pi in)}^k}{D}_{m-k}^{(r)}\right)e^{2\pi inx}\\ =& D_m^{(r)}+\sum _{k=1}^m\frac{{(m)}_k}{k!}{D}_{m-k}^{(r)}(k!\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}^{\mathrm{\infty }}\frac{e^{2\pi inx}}{{(2\pi in)}^k})\\ =& D_m^{(r)}+\sum _{k=2}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-k}^{(r)}{B}_k(\langle x\rangle )+\left(\genfrac{}{}{0ex}{}{m}{1}\right)D_{m-1}^{(r)}\times \{\begin{array}{cc}{B}_1(\langle x\rangle )\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ 0\hfill & \text{for }x\in \mathbb{Z}\hfill \end{array}\\ =& \{\begin{array}{cc}{\sum }_{k=0}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-1}^{(r)}{B}_k(\langle x\rangle )\hfill & \text{for }x\notin \mathbb{Z},\hfill \\ {\sum }_{\begin{array}{c}k=0\\ k\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-1}^{(r)}{B}_k(\langle x\rangle )\hfill & \text{for }x\in \mathbb{Z}.\hfill \end{array}\end{array}$$ 12

Note that (12) holds whether $D_{m-1}^{(r)}=0$ or not. However, if $D_{m-1}^{(r-1)}=0$, then

$$\begin{array}{r}{D}_m^{(r)}(\langle x\rangle )=\sum _{\begin{array}{c}k=0\\ k\ne 1\end{array}}^m\left(\genfrac{}{}{0ex}{}{m}{k}\right)D_{m-1}^{(r)}{B}_k(\langle x\rangle )\text{for all }x\in (-\mathrm{\infty },\mathrm{\infty }).\end{array}$$

Therefore, we obtain the result in Theorem 1.

Assume next that $D_{m-1}^{(r)}\ne 0$. Then we have $D_m^{(r)}(1)\ne D_m^{(r)}$ and hence $D_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and discontinuous with jump discontinuities at integers. Thus the Fourier series of $D_m^{(r)}(\langle x\rangle )$ converges pointwise to $D_m^{(r)}(\langle x\rangle )$ for $x\notin \mathbb{Z}$, and converges to $\frac{1}{2}(D_m^{(r)}+D_m^{(r)}(1))=D_m^{(r)}+(m/2)D_{m-1}^{(r)}$ for $x\in \mathbb{Z}$. Finally, we obtain the formulas in Theorem 2.

From now on we focus on definition (6). Then we can find

$${\mathit{Ch}}_m^{(r)}(x+1)+{\mathit{Ch}}_m^{(r)}(x)=2{\mathit{Ch}}_m^{(r-1)}(x).$$ 13

In other words,

$$\begin{array}{rl}\sum _{m=0}^{\mathrm{\infty }}{\mathit{Ch}}_m^{(r)}(x+1)\frac{t^m}{m!}& ={\left(\frac{2}{2+t}\right)}^r{(1+t)}^{x+1}\\ & =2{\left(\frac{2}{2+t}\right)}^{r-1}{(1+t)}^x-{\left(\frac{2}{2+t}\right)}^r{(1+t)}^x\\ & =2\sum _{m=0}^{\mathrm{\infty }}{\mathit{Ch}}_m^{(r-1)}(x)\frac{t^m}{m!}-\sum _{m=0}^{\mathrm{\infty }}{\mathit{Ch}}_m^{(r)}(x)\frac{t^m}{m!}\\ & =\sum _{m=0}^{\mathrm{\infty }}[2{\mathit{Ch}}_m^{(r-1)}(x)-{\mathit{Ch}}_m^{(r)}(x)]\frac{t^m}{m!}.\end{array}$$

Taking $x=0$ in (13) yields

$$\begin{array}{r}{\mathit{Ch}}_m^{(r)}(1)+{\mathit{Ch}}_m^{(r)}=2{\mathit{Ch}}_m^{(r-1)}(m\ge 0).\end{array}$$

This equation means that

$$\begin{array}{r}{\mathit{Ch}}_m^{(r)}={\mathit{Ch}}_m^{(r)}(1)\iff {\mathit{Ch}}_m^{(r)}={\mathit{Ch}}_m^{(r-1)}.\end{array}$$

Assume that $m\ge 1$ and $r\ge 1$ ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$. In addition, ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ is continuous for those $(r,m)$ with ${\mathit{Ch}}_m^{(r)}={\mathit{Ch}}_m^{(r-1)}$ and discontinuous with jump discontinuities at integers for those $(r,m)$ with ${\mathit{Ch}}_m^{(r)}\ne {\mathit{Ch}}_m^{(r-1)}$. The Fourier series of ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ is

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

Here

$$\begin{array}{rl}{C}_n^{(r,m)}& ={\int }_0^1{\mathit{Ch}}_m^{(r)}(\langle x\rangle )e^{-2\pi inx}dx={\int }_0^1{\mathit{Ch}}_m^{(r)}(x)e^{-2\pi inx}dx\\ & =\frac{1}{m+1}{[{\mathit{Ch}}_{m+1}^{(r)}(x)e^{-2\pi inx}]}_0^1+\frac{2\pi in}{m+1}{\int }_0^1{\mathit{Ch}}_{m+1}^{(r)}(x)e^{-2\pi inx}dx\\ & =\frac{1}{m+1}({\mathit{Ch}}_{m+1}^{(r)}(1)-{\mathit{Ch}}_{m+1}^{(r)})+\frac{2\pi in}{m+1}{C}_n^{(r,m+1)}\\ & =\frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})+\frac{2\pi in}{m+1}{C}_n^{(r,m+1)}.\end{array}$$ 14

By virtue of replacing m by $m-1$ in (14), we can find

$$\begin{array}{r}\frac{2\pi in}{m}{C}_n^{(r,m)}=C_n^{(r,m-1)}+\frac{2}{m}(-{\mathit{Ch}}_m^{(r-1)}+{\mathit{Ch}}_m^{(r)}).\end{array}$$

Case 1

Let $n\ne 0$. Then we acquire that

$$\begin{array}{rl}{C}_n^{(r,m)}=& \frac{m}{2\pi in}{C}_n^{(r,m-1)}+\frac{1}{\pi in}({\mathit{Ch}}_m^{(r)}-{\mathit{Ch}}_m^{(r-1)})\\ =& \frac{m}{2\pi in}(\frac{m-1}{2\pi in}{C}_n^{(r,m-2)}-\frac{1}{\pi in}({\mathit{Ch}}_{m-1}^{(r)}-{\mathit{Ch}}_{m-1}^{(r-1)}))\\ & +\frac{1}{\pi in}({\mathit{Ch}}_m^{(r)}-{\mathit{Ch}}_m^{(r-1)})\\ =& \frac{m(m-1)}{{(2\pi in)}^2}{C}_n^{(r,m-2)}+\frac{m}{2{(\pi in)}^2}({\mathit{Ch}}_{m-1}^{(r)}-{\mathit{Ch}}_{m-1}^{(r-1)})\\ & +\frac{1}{\pi in}({\mathit{Ch}}_m^{(r)}-{\mathit{Ch}}_m^{(r-1)})\\ =& \frac{m(m-1)}{{(2\pi in)}^2}(\frac{m-2}{2\pi in}{C}_n^{(r,m-3)}-\frac{1}{\pi in}({\mathit{Ch}}_{m-2}^{(r)}-{\mathit{Ch}}_{m-2}^{(r-1)}))\\ & +\frac{m}{2{(\pi in)}^2}({\mathit{Ch}}_{m-1}^{(r)}-{\mathit{Ch}}_{m-1}^{(r-1)})+\frac{1}{\pi in}({\mathit{Ch}}_m^{(r)}-{\mathit{Ch}}_m^{(r-1)})\\ =& \frac{m(m-1)(m-2)}{{(2\pi in)}^3}{C}_n^{(r,m-3)}+\frac{m(m-1)}{2^2{(\pi in)}^3}({\mathit{Ch}}_{m-2}^{(r)}-{\mathit{Ch}}_{m-2}^{(r-1)})\\ & +\frac{m}{2{(\pi in)}^2}({\mathit{Ch}}_{m-1}^{(r)}-{\mathit{Ch}}_{m-1}^{(r-1)})+\frac{1}{\pi in}({\mathit{Ch}}_m^{(r)}-{\mathit{Ch}}_m^{(r-1)})\\ =& \cdots \\ =& \frac{m!}{{(2\pi in)}^{m-1}}{C}_n^{(r,1)}+\sum _{k=1}^{m-1}\frac{2{(m)}_k}{{(2\pi in)}^k}({\mathit{Ch}}_{m-k+1}^{(r)}-{\mathit{Ch}}_{m-k+1}^{(r-1)}).\end{array}$$

In addition, we observe that

$$\begin{array}{rl}{C}_n^{(r,1)}& ={\int }_0^1{\mathit{Ch}}_1^{(r)}(x)e^{-2\pi inx}dx={\int }_0^1(x+{\mathit{Ch}}_1^{(r)})e^{-2\pi inx}dx\\ & ={\int }_0^{1}x{e}^{-2\pi inx}dx+{\mathit{Ch}}_1^{(r)}{\int }_0^1{e}^{-2\pi inx}dx\\ & =-\frac{1}{2\pi in}{\left[x{e}^{-2\pi inx}\right]}_0^1+\frac{1}{2\pi in}{\int }_0^1{e}^{-2\pi inx}dx\\ & =-\frac{1}{2\pi in}.\end{array}$$

Therefore, we can derive the following equation:

$$\begin{array}{rl}{C}_n^{(r,m)}& =\frac{-m!}{{(2\pi in)}^m}+\sum _{k=1}^{m-1}\frac{2{(m)}_{k-1}}{{(2\pi in)}^k}({\mathit{Ch}}_{m-k+1}^{(r)}-{\mathit{Ch}}_{m-k+1}^{(r-1)})\\ & =\sum _{k=1}^m\frac{2{(m)}_{k-1}}{{(2\pi in)}^k}({\mathit{Ch}}_{m-k+1}^{(r)}-{\mathit{Ch}}_{m-k+1}^{(r-1)}).\end{array}$$

Here, we used the fact that

$${\mathit{Ch}}_1^{(r)}-{\mathit{Ch}}_1^{(r-1)}=rC{h}_1-(r-1){\mathit{Ch}}_1={\mathit{Ch}}_1=-\frac{1}{2}.$$

Indeed,

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\mathit{Ch}}_n^{(r)}\frac{t^n}{n!}& =\left(\frac{2}{2+t}\right)\times \cdots \times \left(\frac{2}{2+t}\right)\\ & =\sum _{n=0}^{\mathrm{\infty }}(\sum _{l_1+\cdots +l_r=n}\left(\genfrac{}{}{0ex}{}{n}{l_1,l_2,\dots ,l_r}\right){\mathit{Ch}}_{l_1}{\mathit{Ch}}_{l_2}\cdots {\mathit{Ch}}_{l_r})\frac{t^n}{n!}.\end{array}$$

Accordingly, it follows that

$$\begin{array}{rl}{\mathit{Ch}}_1^{(r)}& =\sum _{l_1+\cdots +l_r=1}\left(\genfrac{}{}{0ex}{}{1}{l_1,l_2,\dots ,l_r}\right){\mathit{Ch}}_{l_1}{\mathit{Ch}}_{l_2}\cdots {\mathit{Ch}}_{l_r}\\ & ={\mathit{Ch}}_1+{\mathit{Ch}}_1+\cdots +{\mathit{Ch}}_1=rC{h}_1.\end{array}$$

Case 2

Let $n=0$. Then we have

$$\begin{array}{rl}{C}_0^{(r,m)}& ={\int }_0^1{\mathit{Ch}}_m^{(r)}(x)dx\\ & =\frac{1}{m+1}{[{\mathit{Ch}}_{m+1}^{(r)}(1)-{\mathit{Ch}}_{m+1}^{(r)}]}_0^1\\ & =\frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)}).\end{array}$$

Assume first that ${\mathit{Ch}}_m^{(r)}(1)={\mathit{Ch}}_m^{(r)}$. Then we have ${\mathit{Ch}}_m^{(r)}(1)={\mathit{Ch}}_m^{(r)}$ for $m\ge 2$. ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and continuous. Hence the Fourier series of ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ converges uniformly to ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$, and

$$\begin{array}{rl}{\mathit{Ch}}_m^{(r)}(\langle x\rangle )=& \frac{2}{m+1}({\mathit{Ch}}_{m+1}^{(r-1)}-{\mathit{Ch}}_{m+1}^{(r)})\\ & +\sum _{\begin{array}{c}n=-\mathrm{\infty }\\ n\ne 0\end{array}}\left[\sum _{k=1}^m\frac{2{(m)}_{k-1}}{{(2\pi in)}^k}({\mathit{Ch}}_{m-k+1}^{(r)}-{\mathit{Ch}}_{m-k+1}^{(r-1)})\right]e^{2\pi inx}.\end{array}$$

Consequently, it follows that

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

Thus the proof of Theorem 3 is complete.

Finally, assume that ${\mathit{Ch}}_m^{(r)}\ne {\mathit{Ch}}_m^{(r-1)}$. Then we have ${\mathit{Ch}}_m^{(r)}(1)\ne {\mathit{Ch}}_m^{(r)}$ and hence ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ is piecewise $C^{\mathrm{\infty }}$ and discontinuous with jump discontinuities at integers. Thus the Fourier series of ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ converges pointwise to ${\mathit{Ch}}_m^{(r)}(\langle x\rangle )$ for $x\notin \mathbb{Z}$, and converges to $\frac{1}{2}({\mathit{Ch}}_m^{(r)}+{\mathit{Ch}}_m^{(r)}(1))={\mathit{Ch}}_m^{(r-1)}$ for $x\in \mathbb{Z}$. From the above considerations, the proof of Theorem 4 is complete.

Conclusions

In this paper, the author considered the Fourier series expansion of the higher-order Daehee functions $D_n^{(r)}(\langle x\rangle )$ and the higher-order Changhee functions ${\mathit{Ch}}_n^{(r)}(\langle x\rangle )$ which are obtained by extending by periodicity of period 1 the higher-order Daehee polynomials $D_n^{(r)}(x)$ and the higher-order Changhee polynomials ${\mathit{Ch}}_n^{(r)}(x)$ on $[0,1)$, respectively. The Fourier series are explicitly determined. Depending on whether $D_n^{(r)}(\langle x\rangle )$ and ${\mathit{Ch}}_n^{(r)}(\langle x\rangle )$ are zero or not, the Fourier series of these functions converge uniformly or converge pointwise. In addition, the Fourier series of the higher-order Daehee functions $D_n^{(r)}(\langle x\rangle )$ and the higher-order Changhee functions ${\mathit{Ch}}_n^{(r)}(\langle x\rangle )$ are expressed in terms of the Bernoulli functions $B_k(\langle x\rangle )$. Thus we established the relations between these functions and Bernoulli functions.