Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

Abstract

In this paper, we consider the twisted Carlitz's q-Bernoulli numbers using p-adic q-integral on ℤ _p . From the construction of the twisted Carlitz's q-Bernoulli numbers, we investigate some properties for the twisted Carlitz's q-Bernoulli numbers. Finally, we give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

1. Introduction and preliminaries

Let p be a fixed prime number. Throughout this paper, ℤ _p , ${\mathbb{Q}}_p$ and ${ℂ}_p$ 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 ℕ be the set of natural numbers, and let ℤ₊ = ℕ ∪ {0}. Let ν _p be the normalized exponential valuation of ${ℂ}_p$ with $|p{|}_p=p^{-{\nu }_p\left(p\right)}=\frac{1}{p}$. In this paper, we assume that $q\in {ℂ}_p$ with |1 - q| _p < 1. The q-number is defined by ${\left[x\right]}_q=\frac{1-q^x}{1-q}$. Note that lim_{q → 1}[x] _q = x.

We say that f is a uniformly differentiable function at a point a ∈ ℤ _p , and denote this property by f ∈ UD(ℤ _p ), if the difference quotient $F_f\left(x,y\right)=\frac{f\left(x\right)-f\left(y\right)}{x-y}$ has a limit f'(a) as (x, y) → (a, a). For f ∈ UD(ℤ _p ), the p-adic q-integral on ℤ _p , which is called the q-Volkenborn integral, is defined by Kim as follows:

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

(1)

In [2], Carlitz defined q-Bernoulli numbers, which are called the Carlitz's q-Bernoulli numbers, by

$${\beta }_{0,q}=1,\mathsf{\text{and}}q{\left(q\beta +1\right)}^n-{\beta }_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}n>1,\hfill \end{array}\right.$$

(2)

with the usual convention about replacing βⁿ by β_{n, q}.

In [2, 3], Carlitz also considered the expansion of q-Bernoulli numbers as follows:

$${\beta }_{0,q}^{\left(h\right)}=\frac{h}{{\left[h\right]}_q},\mathsf{\text{and}}{q}^h{\left(q{\beta }^{\left(h\right)}+1\right)}^n-{\beta }_{n,q}^{\left(h\right)}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}n>1,\hfill \end{array}\right.$$

(3)

with the usual convention about replacing (β^(h)) ⁿ by ${\beta }_{n,q}^{\left(h\right)}$.

Let $C_{p^n}=\left\{\xi |{\xi }^{p^n}=1\right\}$ be the cyclic group of order pⁿ , and let $T_p=\underset{n\to \infty }{lim}{C}_{p^n}=C_{p^{\infty }}=\bigcup _{n\ge 0}{C}_{p^n}$ (see [1–16]). Note that T _p is a locally constant space.

For ξ ∈ T _p , the twisted q-Bernoulli numbers are defined by

$$\frac{t}{\xi e^t-1}=e^{B_{\xi }t}=\sum _{n=0}^{\infty }{B}_{n,\xi }\frac{t^n}{n!},$$

(4)

(see [1–19]). From (4), we note that

$$B_{0,q}=0,\mathsf{\text{and}}\xi {\left(B_{\xi }+1\right)}^n-B_{n,\xi }=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}n>1,\hfill \end{array}\right.$$

(5)

with the usual convention about replacing $B_{\xi }^n$ by B _n,ξ (see [17–19]). Recently, several authors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory(see [17–19]).

In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of (3). Using p-adic q-integral equation on ℤ _p , we investigate the properties of the twisted q-Bernoulli numbers and polynomials related to q-Bernstein polynomials. From these properties, we derive some new identities for the twisted q-Bernoulli numbers and polynomials. Final purpose of this paper is to give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

2. On the twisted Carlitz 's q-Bernoulli numbers

In this section, we assume that n ∈ ℤ₊, ξ ∈ T _p and $q\in {ℂ}_p$ with |1 - q| _p < 1.

Let us consider the n th twisted Carlitz's q-Bernoulli polynomials using p-adic q-integral on ℤ _p as follows:

$$\begin{array}{lll}\hfill {\beta }_{n,\xi ,q}\left(x\right)& =\underset{{\mathbb{Z}}_p}{\int }{\left[y+x\right]}_q^n{\xi }^{y}d{\mu }_q\left(y\right)& \hfill \text{(1)}\\ =\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\underset{{\mathbb{Z}}_p}{\int }{\xi }^y{q}^{ly}d{\mu }_q\left(y\right)& \hfill \text{(2)}\\ =\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)\left(\frac{l+1}{1-\xi q^{l+1}}\right){\left(-1\right)}^l{q}^{lx}.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

(6)

In the special case, x = 0, β _n,ξ,q (0) = β _n,ξ,q are called the n th twisted Carlitz's q-Bernoulli numbers.

From (6), we note that

$$\begin{array}{lll}\hfill {\beta }_{n,\xi ,q}\left(x\right)& =\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\left(\frac{1}{1-\xi q^{l+1}}\right)& \hfill \text{(1)}\\ +\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\left(\frac{1}{1-\xi q^{l+1}}\right)& \hfill \text{(2)}\\ =-n\sum _{m=0}^{\infty }{\xi }^m{q}^{2m+x}{\left[x+m\right]}_q^{n-1}+\sum _{m=0}^{\infty }{\xi }^m{q}^m\left(1-q\right){\left[x+m\right]}_q^n.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

(7)

Therefore, by (7), we obtain the following theorem.

Theorem 1. For n ∈ ℤ₊, we have

$${\beta }_{n,\xi ,q}\left(x\right)=-n\sum _{m=0}^{\infty }{\xi }^m{q}^m{\left[x+m\right]}_q^{n-1}+\left(1-q\right)\left(n+1\right)\sum _{m=0}^{\infty }{\xi }^m{q}^m{\left[x+m\right]}_q^n.$$

Let F _{q, ξ} (t, x) be the generating function of the twisted Carlitz's q-Bernoulli poly-nomials, which are given by

$$F_{q,\xi }\left(t,x\right)=e^{{\beta }_{\xi ,q}\left(x\right)t}=\sum _{n=0}^{\infty }{\beta }_{n,\xi ,q}\left(x\right)\frac{t^n}{n!},$$

(8)

with the usual convention about replacing (β _ξ,q (x)) ⁿ by β _n,ξ,q (x).

By (8) and Theorem 1, we get

$$\begin{array}{lll}\hfill F_{q,\xi }\left(t,x\right)& =\sum _{n=0}^{\infty }{\beta }_{n,\xi ,q}\left(x\right)\frac{t^n}{n!}& \hfill \text{(1)}\\ =-t\sum _{m=0}^{\infty }{\xi }^m{q}^{2m+x}{e}^{{\left[x+m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }{\xi }^m{q}^m{e}^{{\left[x+m\right]}_{q}t}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(9)

Let F_q,ξ(t, 0) = F _q,ξ (t). Then, we have

$$q\xi F_{q,\xi }\left(t,1\right)-F_{q,\xi }\left(t\right)=t+\left(q-1\right).$$

(10)

Therefore, by (9) and (10), we obtain the following theorem.

Theorem 2. For n ∈ ℤ₊, we have

$${\beta }_{0,\xi ,q}\left(x\right)=\frac{q-1}{q\xi -1},andq\xi {\beta }_{n,\xi ,q}\left(1\right)-{\beta }_{n,\xi ,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill ifn=1,\hfill \\ \hfill 0\hfill & \hfill ifn>1.\hfill \end{array}\right.$$

From (6), we note that

$$\begin{array}{lll}\hfill {\beta }_{n,\xi ,q}\left(x\right)& =\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_q^{n-l}{q}^{lx}\underset{{\mathbb{Z}}_p}{\int }{\xi }^y{\left[y\right]}_q^{l}d{\mu }_q\left(y\right)& \hfill \text{(1)}\\ =\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_q^{n-l}{q}^{lx}{\beta }_{l,\xi ,q}& \hfill \text{(2)}\\ ={\left({\left[x\right]}_q+q^x{\beta }_{\xi ,q}\right)}^n,& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

(11)

with the usual convention about replacing (β_ξ,q) ⁿ by β_n,ξ,q. By (11) and Theorem 2, we get

$$q\xi {\left(q{\beta }_{\xi ,q}+1\right)}^n-{\beta }_{n,\xi ,q}=\left\{\begin{array}{cc}\hfill q-1\hfill & \hfill \mathsf{\text{if}}n=0,\hfill \\ \hfill 1\hfill & \hfill \mathsf{\text{if}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}n>1.\hfill \end{array}\right.$$

(12)

It is easy to show that

$$\begin{array}{lll}\hfill {\beta }_{n,{\xi }^{-1},q^{-1}}\left(1-x\right)& =\underset{{\mathbb{Z}}_p}{\int }{\xi }^{-y}{\left[1-x+y\right]}_{q^{-1}}^{n}d{\mu }_{q^{-1}}\left(y\right)& \hfill \text{(1) }\\ =\frac{{\left(-1\right)}^n{q}^n}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{-l+lx}\underset{{\mathbb{Z}}_p}{\int }{\xi }^{-y}{q}^{-ly}d{\mu }_{q^{-1}}\left(y\right)& \hfill \text{(2) }\\ =\xi q^n{\left(-1\right)}^n\left(\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\left(\frac{l+1}{1-\xi q^{l+1}}\right)\right)& \hfill \text{(3) }\\ =\xi q^n{\left(-1\right)}^n{\beta }_{n,\xi ,q}\left(x\right).& \hfill \text{(4) }\\ \hfill \text{(5) }\end{array}$$

(13)

Therefore, by (13), we obtain the following theorem.

Theorem 3. For n ∈ ℤ₊, we have

$${\beta }_{n,{\xi }^{-1},q^{-1}}\left(1-x\right)=\xi q^n{\left(-1\right)}^n{\beta }_{n,\xi ,q}\left(x\right).$$

From Theorem 3, we can derive the following functional equation:

$$F_{q^{-1},{\xi }^{-1}}\left(t,1-x\right)=\xi F_{q,\xi }\left(-qt,x\right).$$

(14)

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

Corollary 4. Let$F_{q,\xi }\left(t,x\right)={\sum }_{n=0}^{\infty }{\beta }_{n,\xi ,q}\left(x\right)\frac{t^n}{n!}$. Then we have

$$F_{q^{-1},{\xi }^{-1}}\left(t,1-x\right)=\xi F_{q,\xi }\left(-qt,x\right).$$

By (11), we get that

$$\begin{array}{lll}\hfill q^2{\xi }^2{\beta }_{n,\xi ,q}\left(2\right)& =q^2{\xi }^2\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^l{\left(1+q{\beta }_{\xi ,q}\right)}^l& \hfill \text{(1)}\\ =q^2{\xi }^2\left(\frac{1-q}{1-q\xi }\right)+\left(\begin{array}{c}\hfill n\hfill \\ \hfill 1\hfill \end{array}\right)q^2\xi \left(1+{\beta }_{1,\xi ,q}\right)+q^2{\xi }^2\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^l{\beta }_{l,\xi ,q}\left(1\right)& \hfill \text{(2)}\\ =\left(1-q\right)\frac{q^2{\xi }^2}{1-q\xi }+\left(\begin{array}{c}\hfill n\hfill \\ \hfill 1\hfill \end{array}\right)q^2\xi +q\xi \sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^l{\beta }_{l,\xi ,q}& \hfill \text{(3)}\\ =\frac{1-q}{1-q\xi }{q}^2{\xi }^2+n{q}^2\xi -q\xi \frac{1-q}{1-q\xi }+{\beta }_{n,\xi ,q},\mathsf{\text{if}}n>1.& \hfill \text{(4)}\\ \hfill \text{(5) }\end{array}$$

(15)

Therefore, by (15), we obtain the following theorem.

Theorem 5. For n ∈ ℕ with n > 1, we have

$${\beta }_{n,\xi ,q}\left(2\right)=\frac{1-q}{1-q\xi }+\frac{n}{\xi }-\frac{1}{q\xi }\left(\frac{1-q}{1-q\xi }\right)+{\left(\frac{1}{q\xi }\right)}^2{\beta }_{n,\xi ,q}.$$

By a simple calculation, we easily set

$$\begin{array}{lll}\hfill \xi \underset{{\mathbb{Z}}_p}{\int }{\left[1-x\right]}_{{}^{q^{-1}}}^n{\xi }^{x}d{\mu }_q\left(x\right)& =\xi {\left(-1\right)}^n{q}^n\underset{{\mathbb{Z}}_p}{\int }{\left[x-1\right]}_q^n{\xi }^{x}d{\mu }_q\left(x\right)& \hfill \text{(1)}\\ =\xi {\left(-1\right)}^n{q}^n{\beta }_{n,\xi ,q}\left(-1\right)={\beta }_{n,{\xi }^{-1},q^{-1}}\left(2\right).& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(16)

For n ∈ ℤ₊ with n > 1, we have

$$\begin{array}{lll}\hfill \xi \underset{{\mathbb{Z}}_p}{\int }{\left[1-x\right]}_{q^{-1}}^n{\xi }^{x}d{\mu }_q\left(x\right)& ={\beta }_{n,{\xi }^{-1},q^{-1}}\left(2\right)& \hfill \text{(1)}\\ =\xi \left(\frac{1-q}{1-q\xi }\right)+n\xi -q{\xi }^2\left(\frac{1-q}{1-q\xi }\right)+{\left(q\xi \right)}^2{\beta }_{n,{\xi }^{-1},q^{-1}}& \hfill \text{(2)}\\ =\xi \left(1-q\right)+n\xi +{\left(q\xi \right)}^2{\beta }_{n,{\xi }^{-1},q^{-1}}.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

(17)

Therefore, by (16) and (17), we obtain the following theorem.

Theorem 6. For n ∈ ℤ₊with n > 1, we have

$$\underset{{\mathbb{Z}}_p}{\int }{\left[1-x\right]}_{q^{-1}}^n{\xi }^{x}d{\mu }_q\left(x\right)=\left(1-q\right)+n+q^2\xi {\beta }_{n,{\xi }^{-1},q^{-1}}.$$

For x ∈ℤ _p and n, k ∈ ℤ₊, the p-adic q-Bernstein polynomials are given by

$$B_{k,n}(x,q)=\left(\begin{array}{l}n\hfill \\ k\hfill \end{array}\right){[x]}_q^k{[1-x]}_{q^{-1}}^{n-k},$$

(18)

(see [8, 20]).

In [8], the q-Bernstein operator of order n is given by

$$B_{n,q}\left(f|x\right)=\sum _{k=0}^{n}f\left(\frac{n}{k}\right)B_{k,n}\left(x,q\right)=\sum _{k=0}^{n}f\left(\frac{n}{k}\right)\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_q^k{\left[1-x\right]}_{q^{-1}}^{n-k}.$$

Let f be continuous function on ℤ _p . Then, the sequence $B_{n,q}\left(f|x\right)$ converges uniformly to f on ℤ _p (see [8]). If q is same version in (18), we cannot say that the sequence $B_{n,q}\left(f|x\right)$ converges uniformly to f on ℤ _p .

Let s ∈ ℕ with s ≥ 2. For n₁, ..., n _s , k ∈ ℤ₊ with n₁ + · · · + n _s > sk + 1, we take the p-adic q-integral on ℤ _p for the multiple product of q-Bernstein polynomials as follows:

$$\begin{array}{l}{\displaystyle \underset{{\mathbb{Z}}_p}{\int }{\xi }^x{B}_{k,n_1}(x,q)\cdots B_{k,n_s}(x,q)d{\mu }_q(x)}\\ =\left(\begin{array}{c}{n}_1\\ k\end{array}\right)\dots \left(\begin{array}{c}{n}_s\\ k\end{array}\right){\displaystyle \underset{{\mathbb{Z}}_p}{\int }{[x]}_q^k{[1-x]}_{q^{-1}}^{n_1+\cdots +n_s-sk}{\xi }^{x}d{\mu }_q(x)}\\ =\left(\begin{array}{c}{n}_1\\ k\end{array}\right)\dots \left(\begin{array}{c}{n}_s\\ k\end{array}\right){\displaystyle \sum _{l=0}^{sk}\left(\begin{array}{c}sk\\ l\end{array}\right)}(-1{)}^{l+sk}{\displaystyle \underset{{\mathbb{Z}}_p}{\int }{[1-x]}_{q^{-1}}^{n_1+\cdots +n_s-l}{\xi }^{x}d{\mu }_q(x)}\\ =\left(\begin{array}{l}{n}_1\hfill \\ k\hfill \end{array}\right)\dots \left(\begin{array}{l}{n}_s\hfill \\ k\hfill \end{array}\right){\displaystyle \sum _{l=0}^{sk}\left(\begin{array}{c}sk\\ l\end{array}\right)}(-1{)}^{l+sk}\\ \times (q^2\xi {\beta }_{n_1+\cdots +n_s–l,{\xi }^{–1},q^{–1}}+n_1+\cdots +n_s-l+1-q)d{\mu }_q(x)\\ =\{\begin{array}{ll}{q}^2\xi {\beta }_{n_1+\cdots +n_s,}{}_{{\xi }^{–1},q^{–1}}+n_1+\cdots +n_s+(1-q)\hfill & \text{if}k=0,\hfill \\ q^2\xi \left({}_k^{n_1}\right)\cdots \left({}_k^{n_s}\right){\displaystyle {\sum }_{l=0}^{sk}\left({}_l^{sk}\right)}{(-1)}^{l+sk}{\beta }_{n_1+\cdots +n_s–l,{\xi }^{–1}.q^{–1}}\hfill & \text{if}k>0,\hfill \end{array}\end{array}$$

(19)

and we also have

$$\begin{array}{c}\underset{{\mathbb{Z}}_p}{\int }{\xi }^x{B}_{k,n_1}\left(x,q\right)\cdots B_{k,n_{\mathsf{\text{s}}}}\left(x,q\right)d{\mu }_q\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\dots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n_1+\cdots +n_s-sk}\left(\begin{array}{c}\hfill n_1+\cdots +n_s-sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{\beta }_{l+sk,\xi ,q}.\end{array}$$

(20)

By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.

Theorem 7. Let s ∈ ℕ with s ≥ 2. For n₁, ..., n _s , k ∈ ℤ₊with n₁ + ⋯ + n _s > sk + 1, we have

$$\begin{array}{l}{\displaystyle \sum _{l=0}^{n_1+\cdots +n_s–sk}\left(\begin{array}{c}{n}_1+\cdots +n_s-sk\\ l\end{array}\right)}(-1{)}^l{\beta }_{l+sk,\xi ,q}\\ =\{\begin{array}{ll}{q}^2\xi {\beta }_{n_1+\cdots +n_s,{\xi }^{–1},q^{–1}}+n_1+\cdots +n_s+(1-q)\hfill & ifk=0,\hfill \\ q^2\xi {\displaystyle {\sum }_{l=0}^{sk}\left({}_l^{sk}\right)}{(-1)}^{l+sk}{\beta }_{n_1+\cdots +n_s–l,{\xi }^{–1}.q^{–1}}\hfill & ifk>0.\hfill \end{array}\end{array}$$