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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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