## A note on the weighted $$q$$-Bernoulli numbers and the weighted $$q$$-Bernstein polynomials.(English)Zbl 1243.11011

Let $$p$$ be a fixed prime number and denote $$\mathbb {Z}_p, \mathbb{Q}_p,$$ and $$\mathbb{C}_p$$ as the ring of $$p$$-adic rational integers, the field of $$p$$-adic rational numbers, and the completion of algebraic closure of $$\mathbb{Q}_p$$, respectively. Also, assume that $$\alpha \in \mathbb{Q}$$ and $$q \in \mathbb{C}_p$$ with $$|1-q|_p < p^{- {{1}\over{p-1}}}$$ so that $$q^x = \exp (x \log q).$$ Now define the $$q$$-number $$[x]_q$$ by $$[x]_q = {{1-q^x} \over {1-q}}.$$ Let $$\text{UD}(\mathbb{Z}_p)$$ be the space of uniformly differentiable on $$\mathbb{Z}_p$$ and $$C(\mathbb{Z}_p)$$ be the space of continuous functions on $$\mathbb{Z}_p.$$ For $$f \in C(\mathbb{Z}_p),$$ the weighted $$q$$-Bernstein operator of order $$n$$ for $$f \in \mathbb{Z}_p$$ is defined by $\mathbb B_{n,q}^{(\alpha)} (f) = \sum_{k=0}^{n} f({{k}\over{n}}) \binom{n}{k} [x]_{q^\alpha}^k [1-x]_{q^{-\alpha}}^{n-k} = \sum_{k=0}^{n} f({{k}\over{n}})B_{k,n}^{(\alpha)} (x,q),$ where $B_{k,n}^{(\alpha)} (x,q) = \binom {n}{k} [x]_{q^\alpha}^k [1-x]_{q^{-\alpha}}^{n-k}$ is called the weighted $$q$$-Bernstein polynomials of degree $$n.$$ For $$f \in \text{UD}(\mathbb{Z}_p),$$ the $$p$$-adic $$q$$-integral on $$\mathbb{Z}_p$$ is defined by $I_q(f) = \int_{\mathbb{Z}_p} f(x) d\mu_q (x) = \lim_{N \to \infty} {{1}\over {[p^N]_q}} \sum_{x=0}^{p^N -1} f(x) q^x.$
The modified $$q$$-Bernoulli numbers with weight $$\alpha$$ is defined as ${\tilde{\beta}}_{0,q}^{(\alpha)} = {{q-1} \over{\log q}}, \quad \text{and} \quad (q^{\alpha} {\tilde{\beta}}_{q}^{(\alpha)} + 1 )^n - {\tilde{\beta}}_{n,q}^{(\alpha)} = \begin{cases} {{\alpha}\over{[\alpha]_{q}} } &\text{if $$n=1,$$} \\ 0 &\text{if $$n>1$$} \end{cases}$ with the usual convention about replacing $$( {\tilde{\beta}}_{q}^{(\alpha)})^n$$ by $${\tilde{\beta}}_{n,q}^{(\alpha)}.$$ Also, define the modified $$q$$-Bernoulli polynomials with weight $$\alpha$$ as ${\tilde{\beta}}_{n,q}^{(\alpha)} (x) = \sum_{l=0}^{n} \binom {n}{l} [x]_{q^{\alpha}}^{n-l} q^{\alpha l x } {\tilde{\beta}}_{l,q}^{(\alpha)}.$ In this article, the authors give a $$p$$-adic $$q$$-integral representation on $$\mathbb{Z}_p$$ of the weighted $$q$$-Bernstein polynomials of order $$n$$ associated with the modified $$q$$-Bernoulli numbers and polynomials with weight $$\alpha.$$ As corollaries of these identities they derived some interesting identities on the modified $$q$$-Bernoulli numbers and polynomials with weight $$\alpha.$$

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11B68 Bernoulli and Euler numbers and polynomials 11D88 $$p$$-adic and power series fields 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 05A30 $$q$$-calculus and related topics
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