New q-Euler numbers and polynomials associated with p-adic q-integrals

In this paper we study q-Euler numbers and polynomials by using p-adic q-fermionic integrals on Z_p. The methods to study q-Euler numbers and polynomials in this paper are new.Comment: 13 page

New identities involving q-Euler polynomials of higher order

In this paper we give new identities involving q-Euler polynomials of higher order.Comment: 11 page

A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight alpha and beta via the p-adic q-integral on Zp

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 and q-Genocchi numbers with weight alpha and beta. We apply the method of generating function and p-adic q-integral representation on Zp, 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 on Zp. Also we deduce a fermionic p-adic q-integral representation of product weighted q-Bernstein polynomials of different degrees n1,n2,...on Zp and show that it can be written with q-Genocchi numbers with weight alpha and beta which yields a deeper insight into the effectiveness of this type of generalizations. Our new generating function possess a number of interesting properties which we state in this paper.Comment: 10 page

Observation of inhomogeneous domain nucleation in epitaxial Pb(Zr,Ti)O3 capacitors

We investigated domain nucleation process in epitaxial Pb(Zr,Ti)O3 capacitors under a modified piezoresponse force microscope. We obtained domain evolution images during polarization switching process and observed that domain nucleation occurs at particular sites. This inhomogeneous nucleation process should play an important role in an early stage of switching and under a high electric field. We found that the number of nuclei is linearly proportional to log(switching time), suggesting a broad distribution of activation energies for nucleation. The nucleation sites for a positive bias differ from those for a negative bias, indicating that most nucleation sites are located at ferroelectric/electrode interfaces