A note on [Formula: see text]-Bernstein polynomials and their applications based on [Formula: see text]-calculus

Nowadays [Formula: see text]-Bernstein polynomials have been studied in many different fields such as operator theory, CAGD, and number theory. In order to obtain the fundamental properties and results of Bernstein polynomials by using [Formula: see text]-calculus, we give basic definitions and results related to [Formula: see text]-calculus. The main purpose of this study is to investigate a generating function for [Formula: see text]-Bernstein polynomials. By using an approach similar to that of Goldman et al. in (SIAM J. Discrete Math. 28(3):1009-1025, 2014), we derive some new identities, relations, and formulas for the [Formula: see text]-Bernstein polynomials. Also, we give a plot generating function of [Formula: see text]-Bernstein polynomials for some selected p and q values.