# 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 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, p , ${ℚ}_{p}$ 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 ${ℚ}_{p}$, respectively. Let be the set of natural numbers, and let + = {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 limq → 1[x] q = x.

We say that f is a uniformly differentiable function at a point a p , and denote this property by f UD( 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( p ), the p-adic q-integral on p , which is called the q-Volkenborn integral, is defined by Kim as follows:

(1)

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

${\beta }_{0,q}=1,\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}q{\left(q\beta +1\right)}^{n}-{\beta }_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n>1,\hfill \end{array}\right\$
(2)

with the usual convention about replacing βn by βn, 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}},\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}{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}}\phantom{\rule{1em}{0ex}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}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 [116]). 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 [119]). From (4), we note that

${B}_{0,q}=0,\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\xi {\left({B}_{\xi }+1\right)}^{n}-{B}_{n,\xi }=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n>1,\hfill \end{array}\right\$
(5)

with the usual convention about replacing ${B}_{\xi }^{n}$ by B n,ξ (see [1719]). Recently, several authors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory(see [1719]).

In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of (3). Using p-adic q-integral equation on 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 +, 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 p as follows:

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

$\begin{array}{lll}\hfill {\beta }_{n,\xi ,q}\left(x\right)& =\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\left(\frac{1}{1-\xi {q}^{l+1}}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \phantom{\rule{1em}{0ex}}+\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\left(\frac{1}{1-\xi {q}^{l+1}}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =-n\sum _{m=0}^{\infty }{\xi }^{m}{q}^{2m+x}{\left[x+m\right]}_{q}^{n-1}+\sum _{m=0}^{\infty }{\xi }^{m}{q}^{m}\left(1-q\right){\left[x+m\right]}_{q}^{n}.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(7)

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

Theorem 1. For n +, 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

$\begin{array}{lll}\hfill {F}_{q,\xi }\left(t,x\right)& =\sum _{n=0}^{\infty }{\beta }_{n,\xi ,q}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =-t\sum _{m=0}^{\infty }{\xi }^{m}{q}^{2m+x}{e}^{{\left[x+m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }{\xi }^{m}{q}^{m}{e}^{{\left[x+m\right]}_{q}t}.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(9)

Let Fq,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 +, we have

${\beta }_{0,\xi ,q}\left(x\right)=\frac{q-1}{q\xi -1},\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}q\xi {\beta }_{n,\xi ,q}\left(1\right)-{\beta }_{n,\xi ,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill if\phantom{\rule{1em}{0ex}}n=1,\hfill \\ \hfill 0\hfill & \hfill if\phantom{\rule{1em}{0ex}}n>1.\hfill \end{array}\right\$

From (6), we note that

$\begin{array}{lll}\hfill {\beta }_{n,\xi ,q}\left(x\right)& =\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_{q}^{n-l}{q}^{lx}\underset{{ℤ}_{p}}{\int }{\xi }^{y}{\left[y\right]}_{q}^{l}d{\mu }_{q}\left(y\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_{q}^{n-l}{q}^{lx}{\beta }_{l,\xi ,q}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ ={\left({\left[x\right]}_{q}+{q}^{x}{\beta }_{\xi ,q}\right)}^{n},\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(11)

with the usual convention about replacing (βξ,q) n by βn,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}}\phantom{\rule{1em}{0ex}}n=0,\hfill \\ \hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n=1,\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n>1.\hfill \end{array}\right\$
(12)

It is easy to show that

(13)

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

Theorem 3. For n +, 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

(15)

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

Theorem 5. For 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

$\begin{array}{lll}\hfill \xi \underset{{ℤ}_{p}}{\int }{\left[1-x\right]}_{{}^{{q}^{-1}}}^{n}{\xi }^{x}d{\mu }_{q}\left(x\right)& =\xi {\left(-1\right)}^{n}{q}^{n}\underset{{ℤ}_{p}}{\int }{\left[x-1\right]}_{q}^{n}{\xi }^{x}d{\mu }_{q}\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\xi {\left(-1\right)}^{n}{q}^{n}{\beta }_{n,\xi ,q}\left(-1\right)={\beta }_{n,{\xi }^{-1},{q}^{-1}}\left(2\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(16)

For n + with n > 1, we have

$\begin{array}{lll}\hfill \xi \underset{{ℤ}_{p}}{\int }{\left[1-x\right]}_{{q}^{-1}}^{n}{\xi }^{x}d{\mu }_{q}\left(x\right)& ={\beta }_{n,{\xi }^{-1},{q}^{-1}}\left(2\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\xi \left(\frac{1-q}{1-q\xi }\right)+n\xi -q{\xi }^{2}\left(\frac{1-q}{1-q\xi }\right)+{\left(q\xi \right)}^{2}{\beta }_{n,{\xi }^{-1},{q}^{-1}}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\xi \left(1-q\right)+n\xi +{\left(q\xi \right)}^{2}{\beta }_{n,{\xi }^{-1},{q}^{-1}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(17)

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

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

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

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

${B}_{k,n}\left(x,q\right)=\left(\begin{array}{l}n\hfill \\ k\hfill \end{array}\right){\left[x\right]}_{q}^{k}{\left[1-x\right]}_{{q}^{-1}}^{n-k},$
(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 p . Then, the sequence ${B}_{n,q}\left(f|x\right)$ converges uniformly to f on 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 p .

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

$\begin{array}{l}\underset{{ℤ}_{p}}{\int }{\xi }^{x}{B}_{k,{n}_{1}}\left(x,q\right)\cdots {B}_{k,{n}_{s}}\left(x,q\right)d{\mu }_{q}\left(x\right)\\ =\left(\begin{array}{c}{n}_{1}\\ k\end{array}\right)\dots \left(\begin{array}{c}{n}_{s}\\ k\end{array}\right)\underset{{ℤ}_{p}}{\int }{\left[x\right]}_{q}^{k}{\left[1-x\right]}_{{q}^{-1}}^{{n}_{1}+\cdots +{n}_{s}-sk}{\xi }^{x}d{\mu }_{q}\left(x\right)\\ =\left(\begin{array}{c}{n}_{1}\\ k\end{array}\right)\dots \left(\begin{array}{c}{n}_{s}\\ k\end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}sk\\ l\end{array}\right)\left(-1{\right)}^{l+sk}\underset{{ℤ}_{p}}{\int }{\left[1-x\right]}_{{q}^{-1}}^{{n}_{1}+\cdots +{n}_{s}-l}{\xi }^{x}d{\mu }_{q}\left(x\right)\\ =\left(\begin{array}{l}{n}_{1}\hfill \\ k\hfill \end{array}\right)\dots \left(\begin{array}{l}{n}_{s}\hfill \\ k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}sk\\ l\end{array}\right)\left(-1{\right)}^{l+sk}\\ ×\left({q}^{2}\xi {\beta }_{{n}_{1}+\cdots +{n}_{s}–l,{\xi }^{–1},{q}^{–1}}+{n}_{1}+\cdots +{n}_{s}-l+1-q\right)d{\mu }_{q}\left(x\right)\\ =\left\{\begin{array}{ll}{q}^{2}\xi {\beta }_{{n}_{1}+\cdots +{n}_{s},}{}_{{\xi }^{–1},{q}^{–1}}+{n}_{1}+\cdots +{n}_{s}+\left(1-q\right)\hfill & \text{if}k=0,\hfill \\ {q}^{2}\xi \left({}_{k}^{{n}_{1}}\right)\cdots \left({}_{k}^{{n}_{s}}\right){\sum }_{l=0}^{sk}\left({}_{l}^{sk}\right){\left(-1\right)}^{l+sk}{\beta }_{{n}_{1}+\cdots +{n}_{s}–l,{\xi }^{–1}.{q}^{–1}}\hfill & \text{if}k>0,\hfill \end{array}\end{array}$
(19)

and we also have

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{\xi }^{x}{B}_{k,{n}_{1}}\left(x,q\right)\cdots {B}_{k,{n}_{\mathsf{\text{s}}}}\left(x,q\right)d{\mu }_{q}\left(x\right)\\ \phantom{\rule{1em}{0ex}}=\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\dots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+\cdots +{n}_{s}-sk}\left(\begin{array}{c}\hfill {n}_{1}+\cdots +{n}_{s}-sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{\beta }_{l+sk,\xi ,q}.\end{array}$
(20)

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

Theorem 7. Let s with s ≥ 2. For n1, ..., n s , k +with n1 + + n s > sk + 1, we have

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

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## Acknowledgements

The authors express their sincere gratitude to referees for their valuable suggestions and comments. This paper was supported by the research grant Kwangwoon University in 2011.

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Jang, LC., Kim, T., Kim, YH. et al. Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials. J Inequal Appl 2011, 52 (2011). https://doi.org/10.1186/1029-242X-2011-52