On The Properties Of $q$-Bernstein-Type Polynomials

TL;DR

This paper introduces a new approach to modified q-Bernstein polynomials for multivariable functions, deriving recurrence formulas, identities, generating functions, and exploring their connections with q-analogues of Bernoulli and Euler numbers using p-adic integrals.

Contribution

It presents a novel framework for q-Bernstein polynomials in multiple variables, including new identities, generating functions, and applications to q-analogues of special numbers.

Findings

Derived recurrence formulas and identities for q-Bernstein polynomials.

Established generating functions and interpolation functions for these polynomials.

Connected q-Bernstein polynomials with q-Bernoulli and q-Euler numbers via p-adic integrals.

Abstract

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials.