Mehmet Acikgoz

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.

TL;DR: In this paper, a new generating function of the () Bernstein-type polynomials is proposed, which is based on the Mellin transformation of the first-order Stirling number.

TL;DR: By making use of the generating function methods and summation transform techniques, some new identities are established involving the products of the Apostol-Bernoulli polynomials and the apostol-Genocchi polynomsials.

TL;DR: In this paper, a systemic study of the class of Sheffer sequences in connection with generating function of the weighted q -Euler polynomials is given in the present paper.

TL;DR: In this article, an analogue of Haar distribution based on ( ρ, q ) -numbers is introduced, which is a new generalization of Kim's q-Volkenborn integration defined in [11].