A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials

TL;DR

This paper explores the connections between Bernstein polynomials and Frobenius-Euler numbers and polynomials using generating functions and p-adic integrals, revealing new relations and representations.

Contribution

It introduces new combinatorial relations and integral representations linking Bernstein polynomials with Frobenius-Euler numbers through p-adic analysis.

Findings

Derived combinatorial relations between Frobenius-Euler numbers and polynomials.

Obtained integral representations of Bernstein polynomials on Zp.

Presented fermionic p-adic integral representations of product Bernstein polynomials.

Abstract

The present paper deals with Bernstein polynomials and Frobenius-Euler numbers and polynomials. We apply the method of generating function and fermionic p-adic integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and Frobenius-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between Frobenius-Euler numbers and polynomials. Furthermore, we derive an integral representation of Bernstein polynomials of degree n on Zp . Also we deduce a fermionic p-adic integral representation of product Bernstein polynomials of different degrees n1, n2,...on Zp and show that it can be written with Frobenius-Euler numbers which yields a deeper insight into the effectiveness of this type of generalizations. Our applications possess a number of interesting properties which we state in this paper.