A new family of q-Bernstein polynomials: Probabilistic viewpoint

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these polynomials to establish new relations. These include connections to probabilistic Stirling numbers of the second kind and higher-order probabilistic Bernoulli polynomials associated with Y. Additionally, we derive recurrence and differentiation properties for probabilistic q-Bernstein polynomials. Utilizing Leibniz's formula, we give an identity for the generating function of these polynomials. In the latter part of the paper, we explore applications by choosing appropriate random variables such as Poisson, Bernoulli, Binomial, Geometric, Negative Binomial, and Uniform distributions. This allows us to derive relationships among probabilistic q-Bernstein polynomials, Bell polynomials, Stirling numbers of the second kind, higher-order Frobenius-Euler numbers, and higher-order Bernoulli polynomials. We also present p-adic q-integral and fermionic p-adic q-integral representations for probabilistic q-Bernstein polynomials.