## Some formulae for the $$q$$-Bernstein polynomials and $$q$$-deformed binomial distributions.(English)Zbl 1256.11010

Summary: In this paper, we give new identities involving $$q$$-Bernstein polynomials and we derive some interesting properties of $$q$$-Bernstein polynomials associated with $$q$$-deformed binomial distributions.

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)