A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers
MetadataShow full item record
CitationAraci, S., & Açikgöz, M. (December 01, 2015). A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers. Advances in Difference Equations, 2015, 1.)
The rapid development of q-calculus has led to the discovery of new generalizations of Bernstein polynomials and Genocchi polynomials involving q-integers. The present paper deals with weighted q-Bernstein polynomials (or called q-Bernstein polynomials with weight alpha) and weighted q-Genocchi numbers (or called q-Genocchi numbers with weight alpha and beta). We apply the method of generating function and p-adic q-integral representation on Z(p), which are exploited to derive further classes of Bernstein polynomials and q-Genocchi numbers and polynomials. To be more precise, we summarize our results as follows: we obtain some combinatorial relations between q-Genocchi numbers and polynomials with weight alpha and beta. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n based on Z(p). Also we deduce a fermionic p-adic q-integral representation of products of weighted q-Bernstein polynomials of different degrees n(1), n(2), ... on Z(p) and show that it can be in terms of q-Genocchi numbers with weight alpha and beta, which yields a deeper insight into the effectiveness of this type of generalizations. We derive a new generating function which possesses a number of interesting properties which we state in this paper.
SourceADVANCES IN DIFFERENCE EQUATIONS
Showing items related by title, author, creator and subject.
We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation. © 2014 ...
In this paper, we present a further investigation for the Apostol-Bernoulh polynomials and the Apostol-Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish ...
Motivated by the paper of Kim et al. [T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, D. V. Dolgy, J. Nonlinear Sci. Appl., 9 (2016), 1077-1082], we study a class of q-Euler polynomials earlier given by Kim et al. in [T. Kim, ...