Some identities of special q-polynomials

Abstract

In this paper, we investigate some identities of q-extensions of special polynomials which are derived from the fermonic q-integral on ${\mathbb{Z}}_p$ and the bosonic q-integral on ${\mathbb{Z}}_p$.

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$, and ${\mathbb{C}}_p$ will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ${\mathbb{Q}}_p$, respectively. Let q be an indeterminate in ${\mathbb{C}}_p$ with $|1-q{|}_p<p^{-\frac{1}{p-1}}$ and $UD({\mathbb{Z}}_p)$ be the space of all uniformly differentiable functions on ${\mathbb{Z}}_p$. The q-analog of x is defined as ${[x]}_q=\frac{1-q^x}{1-q}$. Note that ${lim}_{q\to 1}{[x]}_q=x$. For $f\in UD({\mathbb{Z}}_p)$, the bosonic p-adic q-integral on ${\mathbb{Z}}_p$ is defined by Kim to be

$$I_q(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_q(x)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{{[p^N]}_q}\sum _{x=0}^{p^N-1}f(x)q^x(\text{see [1, 2]})$$

(1.1)

and the fermionic p-adic q-integral on ${\mathbb{Z}}_p$ is also defined by Kim to be

$$I_{-q}(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-q}(x)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{{[p^N]}_{-q}}\sum _{x=0}^{p^N-1}f(x){(-q)}^x(\text{see [1–3]}).$$

(1.2)

From (1.1) and (1.2), we have

$$q{I}_q(f_1)-I_q(f)=(q-1)f(0)+\frac{q-1}{logq}{f}^{\prime }(0)$$

(1.3)

and

$$q{I}_{-q}(f_1)+I_{-q}(f)={[2]}_{q}f(0)(\text{see [1–3]}).$$

(1.4)

As is well known, the q-analog of the Bernoulli polynomials is given by the generating function to be

$$\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,q}(x)\frac{t^n}{n!}(\text{see [1, 2, 4–20]}),$$

(1.5)

and the q-analog of the Euler polynomials is given by

$$\frac{{[2]}_q}{q{e}^t+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}(x)\frac{t^n}{n!}(\text{see [1, 2, 4–21]}).$$

(1.6)

The higher-order q-Daehee polynomials are given by

$$\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}(x)\frac{t^n}{n!},$$

(1.7)

where $t\in {\mathbb{C}}_p$ with $|t{|}_p<p^{-\frac{1}{p-1}}$.

Now, we define the q-analog of the Changhee polynomials, which are given by the generating function to be

$$\left(\frac{{[2]}_q}{qt+{[2]}_q}\right){(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{Ch}_{n,q}(x)\frac{t^n}{n!}.$$

(1.8)

In this paper, we investigate some properties for the q-analog of several special polynomials which are derived from the bosonic or fermionic p-adic q-integral on ${\mathbb{Z}}_p$.

2 Some special q-polynomials

In this section, we assume that $t\in {\mathbb{C}}_p$ with $|t{|}_p<p^{-\frac{1}{p-1}}$. Now, we define the higher-order q-Bernoulli numbers,

$${\left(\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,q}^{(r)}(x)\frac{t^n}{n!}.$$

(2.1)

When $x=0$, $B_{n,q}^{(r)}=B_{n,q}^{(r)}(0)$ are called the higher-order q-Bernoulli numbers.

We also consider the higher-order q-Daehee polynomials as follows:

$${\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}^{(r)}(x)\frac{t^n}{n!}.$$

(2.2)

When $x=0$, $D_{n,q}^{(r)}=D_{n,q}^{(r)}(0)$ are called the higher-order q-Daehee numbers.

From (1.3), we can derive the following equation:

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+t)}^{x_1+\cdots +x_r+x}d{\mu }_q(x_1)\cdots d{\mu }_q(x_r)\hfill \\ ={\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{(1+t)}^x\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}^{(r)}(x)\frac{t^n}{n!}.\hfill \end{array}$$

(2.3)

Thus, by (2.3), we get

$${\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{x_1+\cdots +x_r+x}{n}\right)d{\mu }_q(x_1)\cdots d{\mu }_q(x_r)=\frac{D_{n,q}^{(r)}(x)}{n!}(n\ge 0).$$

(2.4)

By replacing t by $e^t-1$ in (2.2), we get

$$\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}^{(r)}(x)\frac{{(e^t-1)}^n}{n!}={\left(\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,q}^{(r)}(x)\frac{t^n}{n!}$$

(2.5)

and

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

(2.6)

Thus, by (2.5) and (2.6), we get

$$B_{n,q}^{(r)}(x)=\sum _{m=0}^n{D}_{m,q}^{(r)}(x)S_2(n,m).$$

(2.7)

Therefore, by (2.4) and (2.7), we obtain the following theorem.

Theorem 1 For $n\ge 0$, we have

$$B_{n,q}^{(r)}(x)=\sum _{m=0}^n{D}_{m,q}^{(r)}(x)S_2(n,m)$$

and

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{x_1+\cdots +x_r+x}{n}\right)d{\mu }_q(x_1)\cdots d{\mu }_q(x_r)\hfill \\ =\frac{D_{n,q}^{(r)}(x)}{n!},\hfill \end{array}$$

where $S_2(n,m)$ is the Stirling number of the second kind.

From (2.1), by replacing t by $log(1+t)$, we obtain

$$\begin{array}{c}{\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{(1+t)}^x\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{B}_{n,q}^{(r)}(x)\frac{1}{n!}{(log(1+t))}^n\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{B}_{n,q}^{(r)}(x)\frac{1}{n!}n!\sum _{m=n}^{\mathrm{\infty }}{S}_1(m,n)\frac{t^m}{m!}\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}(\sum _{n=0}^m{S}_1(m,n)B_{n,q}^{(r)}(x))\frac{t^m}{m!},\hfill \end{array}$$

(2.8)

where $S_1(n,m)$ is the Stirling number of the first kind.

Therefore, by (2.2) and (2.8), we obtain the following theorem.

Theorem 2 For $n\ge 0$, we have

$$D_{n,q}^{(r)}(x)=\sum _{m=0}^n{S}_1(n,m)B_{m,q}^{(r)}(x).$$

Now, we define the higher-order q-Changhee polynomials as follows:

$${\left(\frac{{[2]}_q}{qt+{[2]}_q}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{Ch}_{n,q}^{(r)}(x)\frac{t^n}{n!}.$$

(2.9)

When $x=0$, ${Ch}_{n,q}^{(r)}={Ch}_{n,q}^{(r)}(0)$ are called the higher-order q-Changhee numbers.

From (1.4), we note that

$${\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+t)}^{x_1+\cdots +x_r+x}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_r)={\left(\frac{{[2]}_q}{qt+{[2]}_q}\right)}^r{(1+t)}^x.$$

(2.10)

Thus, by (2.10), we get

$${\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{x_1+\cdots +x_r+x}{n}\right)d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_r)=\frac{{Ch}_{n,q}^{(r)}(x)}{n!}.$$

(2.11)

In view of (1.6), we define the higher-order q-Euler polynomials which are given by the generating function to be

$${\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}^{(r)}(x)\frac{t^n}{n!}.$$

(2.12)

From (2.10), we note that

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+t)}^{x_1+\cdots +x_r+x}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_r)\hfill \\ ={\left(\frac{{[2]}_q}{q{e}^{log(1+t)}+1}\right)}^r{e}^{xlog(1+t)}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}^{(r)}(x)\frac{1}{n!}{(log(1+t))}^n\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}^{(r)}(x)\sum _{m=n}^{\mathrm{\infty }}{S}_1(m,n)\frac{t^m}{m!}\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}(\sum _{n=0}^m{E}_{n,q}^{(r)}(x)S_1(m,n))\frac{t^m}{m!}.\hfill \end{array}$$

(2.13)

Therefore, by (2.11) and (2.13), we obtain the following theorem.

Theorem 3 For $n\ge 0$, we have

$$\begin{array}{r}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{x_1+\cdots +x_r+x}{n}\right)d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_r)\\ =\frac{{Ch}_{n,q}^{(r)}(x)}{n!}=\frac{1}{n!}\sum _{m=0}^n{E}_{m,q}^{(r)}(x)S_1(n,m).\end{array}$$

By replacing t by $e^t-1$ in (2.9), we get

$$\sum _{n=0}^{\mathrm{\infty }}{Ch}_{n,q}^{(r)}(x)\frac{{(e^t-1)}^n}{n!}={\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^r{e}^{xt}$$

(2.14)

and

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{Ch}_{n,q}^{(r)}(x)\frac{1}{n!}{(e^t-1)}^n& =\sum _{n=0}^{\mathrm{\infty }}{Ch}_{n,q}^{(r)}(x)\sum _{m=n}^{\mathrm{\infty }}{S}_2(m,n)\frac{t^m}{m!}\\ =\sum _{m=0}^{\mathrm{\infty }}(\sum _{n=0}^m{Ch}_{n,q}^{(r)}(x)S_2(m,n))\frac{t^m}{n!}.\end{array}$$

(2.15)

Therefore, by (2.12), (2.14), and (2.15), we obtain the following theorem.

Theorem 4 For $m\ge 0$, we have

$$E_{m,q}^{(r)}(x)=\sum _{n=0}^m{Ch}_{n,q}^{(r)}(x)S_2(m,n).$$

Now, we consider the q-analog of the higher-order Cauchy polynomials, which are defined by the generating function to be

$${\left(\frac{q(1+t)-1}{(q-1)+\frac{q-1}{logq}log(1+t)}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{C}_{n,q}^{(r)}(x)\frac{t^n}{n!}.$$

(2.16)

When $x=0$, $C_{n,q}^{(r)}=C_{n,q}^{(r)}(0)$ are called the higher-order q-Cauchy numbers. Indeed,

$$\begin{array}{c}\underset{q\to 1}{lim}{\left(\frac{q(1+t)-1}{q-1+\frac{q-1}{logq}log(1+t)}\right)}^r{(1+t)}^x\hfill \\ ={\left(\frac{t}{log(1+t)}\right)}^r{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(r)}(x)\frac{t^n}{n!},\hfill \end{array}$$

(2.17)

where $C_n^{(r)}(x)$ are called the higher-order Cauchy polynomials.

We observe that

$$\begin{array}{rl}{(1+t)}^x& ={\left(\frac{q(1+t)-1}{q-1+\frac{q-1}{logq}log(1+t)}\right)}^r{(1+t)}^x{\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r\\ =(\sum _{l=0}^{\mathrm{\infty }}{C}_{l,q}^{(r)}(x)\frac{t^l}{l!})\left(\sum _{m=0}^{\mathrm{\infty }}{D}_{m,q}^{(r)}\frac{t^m}{m!}\right)\\ =\sum _{n=0}^{\mathrm{\infty }}(\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)C_{l,q}^{(r)}(x)D_{n-l,q}^{(r)})\frac{t^n}{n!}\end{array}$$

(2.18)

and

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

(2.19)

By (2.18) and (2.19), we get

$${(x)}_n=\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)C_{l,q}^{(r)}(x)D_{n-l,q}^{(r)}.$$

(2.20)

Therefore, by (2.20), we obtain the following theorem.

Theorem 5 For $n\ge 0$, we have

$$\left(\genfrac{}{}{0ex}{}{x}{n}\right)=\frac{1}{n!}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)C_{l,q}^{(r)}(x)D_{n-l,q}^{(r)}.$$

For $n\in \mathbb{N}\cup \{0\}$, we define the q-analog of the Bernoulli-Euler mixed-type polynomials of order $(r,s)$ as follows:

$$B{E}_{n,q}^{(r,s)}(x)={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{E}_{n,q}^{(s)}(x+y_1+\cdots +y_r)d{\mu }_q(y_1)\cdots d{\mu }_q(y_r).$$

(2.21)

Then, by (2.21), we get

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}B{E}_{n,q}^{(r,s)}(x)\frac{t^n}{n!}\hfill \\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\sum _{n=0}^{\mathrm{\infty }}{E}_{n,q}^{(s)}(x+y_1+\cdots +y_r)\frac{t^n}{n!}d{\mu }_q(y_1)\cdots d{\mu }_q(y_r)\hfill \\ ={\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^s{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{e}^{(x+y_1+\cdots +y_r)t}d{\mu }_q(y_1)\cdots d{\mu }_q(y_r)\hfill \\ ={\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^s{\left(\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}\right)}^r{e}^{xt}.\hfill \end{array}$$

(2.22)

It is easy to show that

$${\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^s{\left(\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}(\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)E_{l,q}^{(s)}{B}_{n-l,q}^{(r)}(x))\frac{t^n}{n!}.$$

(2.23)

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 6 For $n\ge 0$, we have

$$B{E}_{n,q}^{(r,s)}(x)=\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)E_{l,q}^{(s)}{B}_{n-l,q}^{(r)}(x).$$

By replacing t by $log(1+t)$ in (2.22), we get

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}B{E}_{n,q}^{(r,s)}(x)\frac{{(log(1+t))}^n}{n!}& ={\left(\frac{{[2]}_q}{qt+{[2]}_q}\right)}^s{\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{(1+t)}^x\\ =\sum _{n=0}^{\mathrm{\infty }}\{\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)D_{m,q}^{(r)}(x){Ch}_{n-m,q}^{(s)}\}\frac{t^n}{n!}\end{array}$$

(2.24)

and

$$\begin{array}{c}\sum _{m=0}^{\mathrm{\infty }}B{E}_{m,q}^{(r,s)}(x)\frac{{(log(1+t))}^m}{m!}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}\{\sum _{m=0}^{n}B{E}_{m,q}^{(r,s)}(x)S_1(n,m)\}\frac{t^n}{n!}.\hfill \end{array}$$

(2.25)

Therefore, by (2.24) and (2.25), we obtain the following theorem.

Theorem 7 For $n\ge 0$, we have

$$\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)D_{m,q}^{(r)}(x){Ch}_{n-m,q}^{(s)}=\sum _{m=0}^{n}B{E}_{m,q}^{(r,s)}(x)S_1(n,m).$$

Let us consider the q-analog of the Daehee-Changhee mixed-type polynomials of order $(r,s)$ as follows: for $n\ge 0$,

$$D{C}_{n,q}^{(r,s)}(x)={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{D}_{n,q}^{(r)}(x+y_1+\cdots +y_s)d{\mu }_{-q}(y_1)\cdots d{\mu }_{-q}(y_s).$$

(2.26)

Thus, by (2.26), we get

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}D{C}_{n,q}^{(r,s)}(x)\frac{t^n}{n!}\hfill \\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}^{(r)}(x+y_1+\cdots +y_s)\frac{t^n}{n!}d{\mu }_{-q}(y_1)\cdots d{\mu }_{-q}(y_s)\hfill \\ ={\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+t)}^{x+y_1+\cdots +y_s}d{\mu }_{-q}(y_1)\cdots d{\mu }_{-q}(y_s)\hfill \\ ={\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^r{\left(\frac{{[2]}_q}{qt+{[2]}_q}\right)}^s{(1+t)}^x\hfill \\ =\left(\sum _{m=0}^{\mathrm{\infty }}{D}_{m,q}^{(r)}\frac{t^m}{m!}\right)(\sum _{l=0}^{\mathrm{\infty }}{Ch}_{l,q}^{(s)}(x)\frac{t^l}{l!})\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}\{\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)D_{m,q}^{(r)}{Ch}_{n-m,q}^{(s)}(x)\}\frac{t^n}{n!}\hfill \end{array}$$

(2.27)

and

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}D{C}_{n,q}^{(r,s)}(x)\frac{{(e^t-1)}^n}{n!}\hfill \\ ={\left(\frac{q-1+\frac{q-1}{logq}t}{q{e}^t-1}\right)}^r{\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^s{e}^{xt}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}\{\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)B_{m,q}^{(r)}{E}_{n-m,q}^{(s)}(x)\}\frac{t^n}{n!}.\hfill \end{array}$$

(2.28)

Now, we observe that

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}D{C}_{n,q}^{(r,s)}(x)\frac{{(e^t-1)}^n}{n!}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}D{C}_{n,q}^{(r,s)}(x)\frac{1}{n!}n!\sum _{m=n}^{\mathrm{\infty }}{S}_2(m,n)\frac{t^m}{m!}\hfill \\ =\sum _{m=0}^{\mathrm{\infty }}\{\sum _{n=0}^{m}D{C}_{n,q}^{(r,s)}(x)S_2(m,n)\}\frac{t^m}{m!}.\hfill \end{array}$$

(2.29)

Therefore, by (2.27), (2.28), and (2.29), we obtain the following theorem.

Theorem 8 For $n\ge 0$, we have

$$D{C}_{n,q}^{(r,s)}(x)=\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)D_{m,q}^{(r)}{Ch}_{n-m,q}^{(s)}(x)$$

and

$$\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)B_{m,q}^{(r)}{E}_{n-m,q}^{(s)}(x)=\sum _{m=0}^{n}D{C}_{m,q}^{(r,s)}(x)S_2(n,m).$$

Now, we consider the q-extension of the Cauchy-Changhee mixed-type polynomials of order $(r,s)$ as follows: for $n\ge 0$,

$$C{C}_{n,q}^{(r,s)}(x)={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{C}_{n,q}^{(r)}(x+y_1+\cdots +y_s)d{\mu }_{-q}(y_1)\cdots d{\mu }_{-q}(y_r).$$

(2.30)

Thus, by (2.30), we get

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}C{C}_{n,q}^{(r,s)}(x)\frac{t^n}{n!}\hfill \\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\sum _{n=0}^{\mathrm{\infty }}{C}_{n,q}^{(r)}(x+y_1+\cdots +y_s)\frac{t^n}{n}d{\mu }_{-q}(y_1)\cdots d{\mu }_{-q}(y_s)\hfill \\ ={\left(\frac{q(1+t)-1}{q-1+\frac{q-1}{logq}log(1+t)}\right)}^r{\left(\frac{{[2]}_q}{qt+{[2]}_q}\right)}^s{(1+t)}^x\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}\{\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)C_{m,q}^{(r)}{Ch}_{n-m,q}^{(s)}(x)\}\frac{t^n}{n!},\hfill \end{array}$$

(2.31)

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}C{C}_{n,q}^{(r,s)}(x)\frac{{(e^t-1)}^n}{n!}\hfill \\ ={\left(\frac{q{e}^t-1}{q-1+\frac{q-1}{logq}t}\right)}^r{\left(\frac{{[2]}_q}{q{e}^t+1}\right)}^s{e}^{tx}\hfill \\ =\left(\sum _{m=0}^{\mathrm{\infty }}{B}_{m,q}^{(-r)}\frac{t^m}{m!}\right)(\sum _{l=0}^{\mathrm{\infty }}{E}_{l,q}^{(s)}(x)\frac{t^l}{l!})\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}(\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)B_{m,q}^{(-r)}{E}_{n-m,q}^{(s)}(x))\frac{t^n}{n!}.\hfill \end{array}$$

(2.32)

Note that

$$\sum _{n=0}^{\mathrm{\infty }}C{C}_{n,q}^{(r,s)}(x)\frac{{(e^t-1)}^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}(\sum _{m=0}^{n}C{C}_{m,q}^{(r,s)}(x)S_2(n,m))\frac{t^n}{n!}.$$

(2.33)

Therefore, by (2.31), (2.32), and (2.33), we obtain the following theorem.

Theorem 9 For $n\ge 0$, we have

$$C{C}_{n,q}^{(r,s)}(x)=\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)C_{m,q}^{(r)}{Ch}_{n-m,q}^{(s)}(x)$$

and

$$\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)B_{m,q}^{(-r)}{E}_{n-m,q}^{(s)}(x)=\sum _{m=0}^{n}C{C}_{m,q}^{(r,s)}(x)S_2(n,m).$$

Finally, we define the q-extension of the Cauchy-Daehee mixed-type polynomials of order $(r,s)$ as follows:

$$C{D}_{n,q}^{(r,s)}(x)={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{C}_{n,q}^{(r)}(x+y_1+\cdots +y_r)d{\mu }_q(x_1)\cdots d{\mu }_q(x_r).$$

(2.34)

Thus, by (2.34), we get

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}C{D}_{n,q}^{(r,s)}(x)\frac{t^n}{n!}\hfill \\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\sum _{n=0}^{\mathrm{\infty }}{C}_{n,q}^{(r)}(x+y_1+\cdots +y_s)\frac{t^n}{n!}d{\mu }_q(y_1)\cdots d{\mu }_q(y_s)\hfill \\ ={\left(\frac{q(1+t)-1}{q-1+\frac{q-1}{logq}log(1+t)}\right)}^r{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+t)}^{x+y_1+\cdots +y_s}d{\mu }_q(y_1)\cdots d{\mu }_q(y_s)\hfill \\ ={\left(\frac{q(1+t)-1}{q-1+\frac{q-1}{logq}log(1+t)}\right)}^r{\left(\frac{q-1+\frac{q-1}{logq}log(1+t)}{q(1+t)-1}\right)}^s{(1+t)}^x\hfill \\ =\{\begin{array}{cc}{\sum }_{n=0}^{\mathrm{\infty }}{C}_{n,q}^{(r-s)}(x)\frac{t^n}{n!}\hfill & \text{if }r>s,\hfill \\ {\sum }_{n=0}^{\mathrm{\infty }}{D}_{n,q}^{(s-r)}(x)\frac{t^n}{n!}\hfill & \text{if }r<s,\hfill \\ {\sum }_{n=0}^{\mathrm{\infty }}{(x)}_n\frac{t^n}{n!}\hfill & \text{if }r=s.\hfill \end{array}\hfill \end{array}$$

(2.35)

Therefore, by (2.35), we obtain the following equation:

$$C{D}_{n,q}^{(r,s)}(x)=\{\begin{array}{cc}{C}_{n,q}^{(r-s)}(x)\hfill & \text{if }r>s,\hfill \\ D_{n,q}^{(s-r)}(x)\hfill & \text{if }r<s,\hfill \\ {(x)}_n\hfill & \text{if }r=s.\hfill \end{array}$$