Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Some identities on the weighted q-Euler numbers and q-Bernstein polynomials | Kim et al. Journal of Inequalities and Applications 2011 2011 64 http content 2011 1 64 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some identities on the weighted q-Euler numbers and q-Bernstein polynomials Taekyun Kim 1 Young-Hee Kim1 and Cheon S Ryoo2 Correspondence tkkim@ 1Division of General Education- Mathematics Kwangwoon University Seoul 139-701 Korea Full list of author information is available at the end of the article Springer Abstract Recently Ryoo introduced the weighted q-Euler numbers and polynomials which are a slightly different Kim s weighted q-Euler numbers and polynomials see C. S. Ryoo A note on the weighted q-Euler numbers and polynomials 2011 . In this paper we give some interesting new identities on the weighted q-Euler numbers related to the q-Bernstein polynomials 2000 Mathematics Subject Classification - 11B68 11S40 11S80 Keywords Euler numbers and polynomials q-Euler numbers and polynomials weighted q-Euler numbers and polynomials Bernstein polynomials q-Bernstein polynomials 1. Introduction Let p be a fixed odd prime number. Throughout this paper zp Qp c and cp will denote the ring of Ji-adic integers the field of Ji-adic rational numbers the complex number fields and the completion of algebraic closure of Qp respectively. Let N be the set of natural numbers and Z N u 0 . Let vp be the normalized exponential valuation of cp with IpIp p p p p. When one talks of q-extension q is variously considered as an indeterminate a complex number q e c or a Ji-adic number q e cp. If q e c then one normally assumes q 1 and if q e cp then one normally assumes q - 1 p 1. In this paper the q-number is defined by 1 - 4 I q see 1-19 Note that limq 1 x q x see 1-19 . Let f be a continuous function on zp. For a e N and k n e z the weighted Ji-adic q-Bernstein operator of order n for f is defined by Kim as follows roWffIr Y n f k mA II vi k Bn q V x y I k f n x a 1
