Some Identities on the Twisted q-Analogues of Catalan-Daehee Numbers and Polynomials

Abstract

In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on ${\mathbb{Z}}_p$. We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper, ${\mathbb{Z}}_p,{\mathbb{Q}}_p$ and ${\mathbb{C}}_p$ we denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ${\mathbb{Q}}_p$. The p-adic norm ${|·|}_p$ is normally defined ${\left|p\right|}_p=\frac{1}{p}$. Let q be an indeterminate in ${\mathbb{C}}_p$ with ${|1-q|}_p<p^{-\frac{1}{p-1}}$. The q-analogue of x is defined by ${\displaystyle {\left[x\right]}_q=\frac{1-q^x}{1-q}}$. Note that ${\displaystyle \underset{q\to 1}{lim}{\left[x\right]}_q=x.}$

Let $f\left(x\right)$ be a uniformly differentiable function on ${\mathbb{Z}}_p$. Then the p-adic q-integral on ${\mathbb{Z}}_p$ is defined by [1,2,3]

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_p}f\left(x\right)d{\mu }_q\left(x\right)& =\underset{N\to \infty }{lim}\sum _{x=0}^{p^N-1}f\left(x\right){\mu }_q(x+p^N{\mathbb{Z}}_p)\hfill \\ \hfill & =\underset{N\to \infty }{lim}\frac{1}{{\left[p^N\right]}_q}\sum _{x=0}^{p^N-1}f\left(x\right)q^x.\hfill \end{array}$$

(1)

From (1), we have

$$q{\int }_{{\mathbb{Z}}_p}f(x+1)d{\mu }_q\left(x\right)={\int }_{{\mathbb{Z}}_p}f\left(x\right)d{\mu }_q\left(x\right)+(q-1)f\left(0\right)+\frac{q-1}{logq}{f}^{\prime }\left(0\right),$$

(2)

where $f^{\prime }\left(0\right)=\frac{df\left(x\right)}{d}{|}_{x=0}$.

For $n\in \mathbb{N}$, let $T_p$ be the p-adic locally constant space defined by

$$\begin{array}{c}\hfill T_p=\bigcup _{n\ge 1}{C}_{p^n}=\underset{n\to \infty }{lim}{C}_{p^n},\end{array}$$

where $C_{p^n}=\left\{w\right|w^{p^n}=1\}$ is the cyclic group of order $p^n$.

For $w\in T_p$, let us take $f\left(x\right)=w^x{e}^{xt}$. Then, by (1), we get

$$\frac{(q-1)+\frac{q-1}{logq}t}{wq{e}^t-1}={\int }_{{\mathbb{Z}}_p}{w}^x{e}^{xt}d{\mu }_q\left(x\right)$$

(3)

Thus, by (3), we define the twisted q-Bernoulli numbers which are given by the generating function to be

$$\frac{(q-1)+\frac{q-1}{logq}t}{qw{e}^t-1}=\sum _{n=0}^{\infty }{B}_{n,q,w}\frac{t^n}{n!}.$$

(4)

From (4), we note that

$$qw{(B_{q,w}+1)}^n-B_{n,q,w}=\left\{\begin{array}{cc}q-1,& \mathrm{if}n=0\\ {\displaystyle \frac{q-1}{logq}}& \mathrm{if}n=1,\\ 0& \mathrm{if}n\ge 1,\end{array}\right.$$

with the usual convention about replacing $B_{q,w}^n$ by $B_{n,q,w}$.

From (2) and (4), we have

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\int }_{{\mathbb{Z}}_p}{w}^x{x}^{n}d{\mu }_q\left(x\right)\frac{t^n}{n!}& ={\int }_{{\mathbb{Z}}_p}{w}^x{e}^{xt}d{\mu }_q\left(x\right)\hfill \\ \hfill & =\frac{(q-1)+\frac{q-1}{logq}t}{wq{e}^t-1}=\sum _{n=0}^{\infty }{B}_{n,q,w}\frac{t^n}{n!}.\hfill \end{array}$$

(5)

Thus, by (5), we get

$${\int }_{{\mathbb{Z}}_p}{w}^x{x}^{n}d{\mu }_q\left(x\right)=B_{n,q,w},(n\ge 0).$$

(6)

For ${\left|t\right|}_p<p^{-\frac{1}{p-1}}$, the twisted $(\lambda ,q)$-Daehee polynomials are defined by generating function to be (cf. [4])

$$\sum _{n=0}^{\infty }{D}_{n,q,w}\left(x\right|\lambda )\frac{t^n}{n!}=\frac{2(q-1)+\lambda \frac{q-1}{logq}log(1+t)}{w{q}^2{(1+t)}^{\lambda }-1}{(1+t)}^{\lambda x}.$$

(7)

When $x=0$, $D_{n,q,w}\left(\lambda \right)=D_{n,q,w}\left(0\right|\lambda )$ are called the twisted $(\lambda ,q)$-Daehee numbers. In particular,

$$D_{0,q,w}\left(1\right)=\frac{2(q-1)}{w{q}^2-1}.$$

The twisted Catalan-Daehee numbers are defined by [5]

$$\frac{\frac{1}{2}log(1-4t)}{w\sqrt{1-4t}-1}=\sum _{n=0}^{\infty }{d}_{n,w}{t}^n.$$

(8)

If we take $w=1$ in the twisted Catalan-Daehee numbers, $d_n=d_{n,1}$, are the Catalan-Daehee numbers in [6,7,8].

We note that

$$\sqrt{1+t}=\sum _{m=0}^{\infty }{(-1)}^{m-1}\left(\genfrac{}{}{0pt}{}{2m}{m}\right){\left(\frac{1}{4}\right)}^m\left(\frac{1}{2m-1}\right)t^m.$$

(9)

By replacing t by $-4t$ in (9), we get

$$\sqrt{1-4t}=1-2\sum _{m=0}^{\infty }\left(\genfrac{}{}{0pt}{}{2m}{m}\right)\frac{1}{m+1}{t}^{m+1}=1-2\sum _{m=0}^{\infty }{C}_m{t}^{m+1},$$

(10)

where $C_m$ is the Catalan number.

From (8) and (10), Dolgy et al. showed a relation between the Catalan-Daehee numbers and the Catalan numbers in [6];

$$d_n=\left\{\begin{array}{cc}1,& \mathrm{if}n=0\\ {\displaystyle \frac{4^n}{n+1}-\sum _{m=0}^{n-1}\frac{4^{n-m-1}}{n-m}{C}_m,}& \mathrm{if}n\ge 1.\end{array}\right.$$

Catalan-Daehee numbers and polynomials were introduced in [7] and considered the family of linear differential equations arising from the generating function of those numbers in order to derive some explicit identities involving Catalan-Daehee numbers and Catalan numbers. In [8], several properties and identities associated with Catalan-Daehee numbers and polynomials were derived by utilizing umbral calculus techniques. Dolgy et al. gave some new identities for those numbers and polynomials derived from p-adic Volkenborn integrals on ${\mathbb{Z}}_p$ in [6]. Recently, Ma et al. introduced and studied q-analogues of the Catalan-Daehee numbers and polynomials with the help of p-adic q-integral on ${\mathbb{Z}}_p$ in [9]. The aim of this paper is to introduce q-analogues of the twisted Catalan-Daehee numbers and polynomials by using p-adic q-integral on ${\mathbb{Z}}_p$, and derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials.

2. The Twisted $\mathit{Q}$-Analogues of Catalan-Daehee Numbers

For $t\in {\mathbb{C}}_p$ with ${\left|t\right|}_p<p^{-\frac{1}{p-1}}$ and for $w\in T_p$, we have

$${\int }_{{\mathbb{Z}}_p}{w}^x{(1-4t)}^{\frac{x}{2}}d{\mu }_q\left(x\right)=\frac{q-1+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{qw\sqrt{1-4t}-1}.$$

(11)

In the view of (11), we define the twisted q-analogue of Catalan-Daehee numbers which are given by the generating function to be

$$\frac{q-1+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{qw\sqrt{1-4t}-1}=\sum _{n=0}^{\infty }{d}_{n,q,w}{t}^n.$$

(12)

Note that ${\displaystyle \underset{q\to 1}{lim}{d}_{n,q,w}=d_{n,w},(n\ge 0)}$, which is the twisted Catalan-Daehee numbers in [5].

From (7) and (12), we have

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{d}_{n,q,w}{t}^n& =\frac{1}{2}\left(\frac{2(q-1)+\frac{q-1}{logq}log(1-4t)}{w^2{q}^2(1-4t)-1}\right)\left(qw\sqrt{1-4t}+1\right)\hfill \\ \hfill & =\frac{1}{2}\left(\sum _{l=0}^{\infty }{4}^l{D}_{l,q,w^2}\left(1\right)\frac{{(-t)}^l}{l!}\right)\left(1+qw-2qw\sum _{m=0}^{\infty }{C}_m{t}^{m+1}\right)\hfill \\ \hfill & =\frac{1+qw}{2}\sum _{n=0}^{\infty }{(-4)}^n\frac{D_{n,q,w}\left(1\right)}{n!}{t}^n-qw\sum _{n=1}^{\infty }\left(\sum _{m=0}^{n-1}\frac{{(-4)}^{n-m-1}}{(n-m-1)!}{D}_{n-m-1,q,w^2}\left(1\right)C_m\right)t^n\hfill \\ \hfill & =\frac{q^2-1}{w{q}^2-1}+\sum _{n=0}^{\infty }\frac{{\left[2\right]}_{qw}}{2}{(-4)}^n\frac{D_{n,q,w}\left(1\right)}{n!}{t}^n-qw\sum _{n=1}^{\infty }\left(\sum _{m=0}^{n-1}\frac{{(-4)}^{n-m-1}}{(n-m-1)!}{D}_{n-m-1,q,w^2}\left(1\right)C_m\right)t^n\hfill \\ \hfill & =\frac{q^2-1}{w{q}^2-1}+\sum _{n=1}^{\infty }\left(\frac{{\left[2\right]}_{qw}}{2}\frac{{(-4)}^n}{n!}{D}_{n,q,w^2}\left(1\right)-qw\sum _{m=0}^{n-1}\frac{{(-4)}^{n-m-1}}{(n-m-1)!}{D}_{n-m-1,q,w^2}\left(1\right)C_m\right)t^n.\hfill \end{array}$$

(13)

Therefore, by comparing the coefficients on the both sides of (13), we obtain the following theorem.

Theorem 1.

For $n\ge 0$ and $w\in T_p$, we have

$$d_{n,q,w}=\left\{\begin{array}{cc}{\displaystyle \frac{q^2-1}{w{q}^2-1}},& \mathrm{if}n=0,\\ {\displaystyle \frac{1+qw}{2}\frac{{(-4)}^n}{n!}{D}_{n,q,w^2}\left(1\right)-qw\sum _{m=0}^{n-1}\frac{{(-4)}^{n-m-1}}{(n-m-1)!}{2}^{2n-2m-1}{D}_{n-m-1,q,w^2}\left(1\right)C_m,}& \mathrm{if}n\ge 1.\end{array}\right.$$

Specially, $w=1$ and $q\to 1$, we have

Corollary 1

(Theorem 1, [6]). For $n\ge 0$, we have

$$d_n=\left\{\begin{array}{cc}1,& \mathrm{if}n=0,\\ {\displaystyle {(-4)}^n\frac{D_n\left(1\right)}{n!}-\sum _{m=0}^{n-1}\frac{{(-4)}^{n-m-1}}{(n-m-1)!}{2}^{2n-2m-1}{D}_{n-m-1}\left(1\right)C_m,}& \mathrm{if}n\ge 1.\end{array}\right.$$

Now, from (6) and (12), we observe that

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{d}_{n,q,w}{t}^n& =\frac{q-1+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{qw\sqrt{1-4t}-1}={\int }_{{\mathbb{Z}}_p}{w}^x{(1-4t)}^{\frac{x}{2}}d{\mu }_q\left(x\right)\hfill \\ \hfill & =\sum _{m=0}^{\infty }{\left(\frac{1}{2}\right)}^m\frac{1}{m!}{\left(log(1-4t)\right)}^m{\int }_{{\mathbb{Z}}_p}{w}^x{x}^{m}d{\mu }_q\left(x\right)\hfill \\ \hfill & =\sum _{m=0}^{\infty }{\left(\frac{1}{2}\right)}^m{B}_{m,q,w}\sum _{n=m}^{\infty }{S}_1(n,m)\frac{1}{n!}{(-4t)}^n\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left(\sum _{m=0}^n{2}^{2n-m}{(-1)}^n{B}_{m,q,w}{S}_1(n,m)\right)\frac{t^n}{n!},\hfill \end{array}$$

(14)

where $S_1(n,m),(n,m\ge 0)$ is the Stirling number of the first kind which is defined by [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]

$${\left(x\right)}_n=\sum _{l=0}^n{S}_1(n,l)x^l,(n\ge 0).$$

Here, ${\left(x\right)}_0=1$, ${\left(x\right)}_n=x(x-1)\cdots (x-n+1)$, $(n\ge 1)$.

Therefore, by (14), we obtain the following theorem.

Theorem 2.

For $n\ge 0$ and $w\in T_p$, we have

$${(-1)}^n{d}_{n,q,w}=\frac{1}{n!}\sum _{m=0}^n{2}^{2n-m}{B}_{m,q,w}{S}_1(n,m).$$

By binomial expansion, we get

$${\int }_{{\mathbb{Z}}_p}{w}^x{(1-4t)}^{\frac{x}{2}}d{\mu }_q\left(x\right)=\sum _{n=0}^{\infty }{(-4)}^n{\int }_{{\mathbb{Z}}_p}{w}^x\left(\genfrac{}{}{0pt}{}{\frac{x}{2}}{n}\right)d{\mu }_q\left(x\right)t^n.$$

(15)

From (12) and (15), we obtain the following corollary.

Corollary 2.

For $n\ge 0$ and $w\in T_p$, we have

$${\int }_{{\mathbb{Z}}_p}{w}^x\left(\genfrac{}{}{0pt}{}{\frac{x}{2}}{n}\right)d{\mu }_q\left(x\right)={(-1)}^n{2}^{-2n}{d}_{n,q,w}=\frac{1}{n!}\sum _{m=0}^n{\left(\frac{1}{2}\right)}^m{B}_{m,q,w}{S}_1(n,m).$$

For the case $w=1$ and $q\to 1$, we have the following.

Corollary 3

(Theorem 2, [6]). For $n\ge 0$, we have

$${(-1)}^n{d}_n=\frac{1}{n!}\sum _{m=0}^n{2}^{2n-m}{B}_m{S}_1(n,m).$$

The twisted q-analogue of $\lambda$-Daehee polynomials are given by the p-adic q-integral on ${\mathbb{Z}}_p$ to be

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_p}{w}^y{(1+t)}^{\lambda y+x}d{\mu }_q\left(y\right)& =\frac{(q-1)+\lambda \frac{q-1}{logq}log(1+t)}{qw{(1+t)}^{\lambda }-1}{(1+t)}^x\hfill \\ \hfill & =\sum _{n=0}^{\infty }{\tilde{D}}_{n,q,w}\left(x\right|\lambda )\frac{t^n}{n!}.\hfill \end{array}$$

(16)

When $x=0$, ${\tilde{D}}_{n,q,w}\left(\lambda \right)={\tilde{D}}_{n,q,w}\left(0\right|\lambda )(n\ge 0)$ are called the twisted q-analogue of $\lambda$-Daehee numbers. Note that

$${\tilde{D}}_{0,q,w}\left(\lambda \right)=\frac{q-1}{wq-1}.$$

From (16), we note that

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{(-1)}^n{4}^n{\tilde{D}}_{n,q,w}\left(\frac{1}{2}\right)\frac{t^n}{n!}& =\frac{q-1+\frac{1}{2}\frac{q-1}{logq}log(1-4t)}{qw{(1-4t)}^{\frac{1}{2}}-1}\hfill \\ \hfill & =\sum _{n=0}^{\infty }{d}_{n,q,w}{t}^n.\hfill \end{array}$$

(17)

Thus, by (17), we get

$$d_{n,q,w}={(-1)}^n\frac{4^n}{n!}{\tilde{D}}_{n,q,w}\left(\frac{1}{2}\right),(n\ge 0).$$

Let us take $t=\frac{1}{4}(1-e^{2t})$ in (12). Then we have

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{d}_{k,q,w}{\left(\frac{1}{4}\right)}^k{(1-e^{2t})}^k& =\frac{q-1+\frac{q-1}{logq}t}{qw{e}^t-1}={\int }_{{\mathbb{Z}}_p}{w}^x{e}^{xt}d{\mu }_q\left(x\right)\hfill \\ \hfill & =\sum _{n=0}^{\infty }{B}_{n,q,w}\frac{t^n}{n!}.\hfill \end{array}$$

(18)

On the other hand,

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{d}_{k,q,w}{(-1)}^k{\left(\frac{1}{4}\right)}^k{(e^{2t}-1)}^k& =\sum _{k=0}^{\infty }{(-1)}^{k}k!d_{k,q,w}{\left(\frac{1}{4}\right)}^k\frac{1}{k!}{\left(e^{2t}-1\right)}^k\hfill \\ \hfill & =\sum _{k=0}^{\infty }{(-1)}^{k}k!d_{k,q,w}{2}^{-2k}\sum _{n=k}^{\infty }{S}_2(n,k)2^n\frac{t^n}{n!}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left(\sum _{k=0}^n{(-1)}^{k}k!d_{k,q,w}{2}^{n-2k}{S}_2(n,k)\right)\frac{t^n}{n!},\hfill \end{array}$$

(19)

where $S_2(n,k)(n,k\ge 0)$ is the Stirling number of the second kind which is defined by

$$x^n=\sum _{l=0}^n{S}_2(n,l){\left(x\right)}_l,(n\ge 0).$$

Therefore, by (18) and (19), we obtain the following theorem.

Theorem 3.

For $n\ge 0$, we have

$$B_{n,q,w}=\sum _{k=0}^n{(-1)}^k{S}_2(n,k)2^{n-2k}k!d_{k,q,w}.$$

Now, we observe that

$${\int }_{{\mathbb{Z}}_p}{w}^y{(1-4t)}^{\frac{x+y}{2}}d{\mu }_q\left(y\right)=\frac{(q-1)+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{wq\sqrt{1-4t}-1}{(1-4t)}^{\frac{x}{2}}.$$

We define the twisted Catalan-Daehee polynomials which are given by the generating function to be

$$\frac{q-1+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{qw\sqrt{1-4t}-1}{(1-4t)}^{\frac{x}{2}}=\sum _{n=0}^{\infty }{d}_{n,q,w}\left(x\right)t^n.$$

(20)

When $x=0$, $d_{n,q,w}=d_{n,q,w}\left(0\right)$$(n\ge 0)$ are the twisted Catalan-Daehee numbers in (12).

Note that

$$\begin{array}{cc}\hfill {(1-4t)}^{\frac{x}{2}}& =\sum _{l=0}^{\infty }{\left(\frac{x}{2}\right)}^l\frac{1}{l!}{\left(log(1-4t)\right)}^l=\sum _{l=0}^{\infty }{\left(\frac{x}{2}\right)}^l\sum _{m=l}^{\infty }{S}_1(m,l){(-4)}^m\frac{t^m}{m!}\hfill \\ \hfill & =\sum _{m=0}^{\infty }\left(\sum _{l=0}^m{S}_1(m,l)\frac{{(-4)}^m}{m!}{\left(\frac{x}{2}\right)}^l\right)t^m.\hfill \end{array}$$

(21)

Thus, by (12), (20) and (21), we get

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{d}_{n,q,w}\left(x\right)t^n& =\frac{q-1+\frac{q-1}{logq}\frac{1}{2}log(1-4t)}{qw\sqrt{1-4t}-1}{(1-4t)}^{\frac{x}{2}}\hfill \\ \hfill & =\left(\sum _{k=0}^{\infty }{d}_{k,q,w}\frac{t^k}{k!}\right)\left(\sum _{m=0}^{\infty }\sum _{l=0}^m{S}_1(m,l)\frac{{(-4)}^m}{m!}{\left(\frac{x}{2}\right)}^l\right)t^m\hfill \\ \hfill & =\sum _{n=0}^{\infty }\left(\sum _{m=0}^n\sum _{l=0}^m{S}_1(m,l)\frac{{(-4)}^m}{m!}{d}_{n-m,q,w}{\left(\frac{x}{2}\right)}^l\right)t^n.\hfill \end{array}$$

(22)

By comparing the coefficients on the both sides (22), we obtain the following theorem.

Theorem 4.

For $n\ge 0$, we have

$$\begin{array}{cc}\hfill d_{n,q,w}\left(x\right)& =\sum _{m=0}^n\sum _{l=0}^m{S}_1(m,l){(-1)}^m\frac{2^{2m-l}}{m!}{d}_{n-m,q,w}{x}^l\hfill \\ \hfill & =\sum _{l=0}^n\left(\sum _{m=l}^n{(-1)}^m\frac{2^{2m-l}}{m!}{S}_1(m,l)d_{n-m,q,w}\right)x^l.\hfill \end{array}$$

For the case $w=1$ and $q\to 1$, we have the following.

Corollary 4

(Theorem 5, [6]). For $n\ge 0$, we have

$$\begin{array}{cc}\hfill d_n\left(x\right)& =\sum _{l=0}^n\left(\sum _{m=l}^n{(-1)}^m\frac{2^{2m-l}}{m!}{S}_1(m,l)d_{n-m}\right)x^l.\hfill \end{array}$$

3. Conclusions

To summarize, we introduced twisted q-analogues of Catalan-Daehee numbers and polynomials and obtained several explicit expressions and identities related to them. We expressed the twisted q-analogues of Catalan-Daehee numbers in terms of the twisted $(\lambda ,q)$-Daehee numbers, and of the twisted q-Bernoulli numbers and Stirling numbers of the first kind in Theorems 1 and 2. We also derived an identity involving the twisted q-Bernoulli numbers, twisted q-analogues of Catalan-Daehee numbers and Stirling numbers of the second kind in Theorem 3. In addition, we obtain an explicit expression for the twisted q-analogues of Catalan-Daehee polynomials which involve the twisted q-analogues of Catalan-Daehee numbers and Stirling numbers of the first kind in Theorem 4.

In recent years, many special numbers and polynomials have been studied by employing various methods, including: generating functions, p-adic analysis, combinatorial methods, umbral calculus, differential equations, probability theory and analytic number theory. We are now interested in continuing our research into the application of ‘twisted’ and ‘q-analogue’ versions of certain interesting special polynomials and numbers in the fields of physics, science, and engineering as well as mathematics.