\(q\)-Bernstein polynomials associated with \(q\)-Stirling numbers and Carlitz’s \(q\)-Bernoulli numbers. (English) Zbl 1208.11029

Kim, T.; Choi, J.; Kim, Y. H.

Abstr. Appl. Anal. 2010, Article ID 150975, 11 p. (2010).

Summary: Recently, the first author [Russ. J. Math. Phys. 18, No. 1, 73–82 (2011; Zbl 1256.11018)] introduced \(q\)-Bernstein polynomials which are different from G. M. Phillips [Ann. Numer. Math. 4, No. 1–4, 511–518 (1997; Zbl 0881.41008)]. In this paper, we give a \(p\)-adic \(q\)-integral representation for \(q\)-Bernstein type polynomials and investigate some interesting identities of \(q\)-Bernstein type polynomials associated with \(q\)-extensions of the binomial distribution, \(q\)-Stirling numbers, and Carlitz’s \(q\)-Bernoulli numbers.

MSC:

11B68	Bernoulli and Euler numbers and polynomials
11B65	Binomial coefficients; factorials; \(q\)-identities
11B73	Bell and Stirling numbers
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Citations:

Zbl 0881.41008; Zbl 1256.11018

References:

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