Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A new generalization of the Riemann zeta function and its difference equation | Chaudhry et al. Advances in Difference Equations 2011 2011 20 http content 2011 1 20 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access A new generalization of the Riemann zeta function and its difference equation Muhammad Aslam Chaudhry 1 Asghar Qadir2 and Asifa Tassaddiq2 Correspondence maslam@kfupm. department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia Full list of author information is available at the end of the article SpringerOpen0 Abstract We have introduced a new generalization of the Riemann zeta function. A special case of our generalization converges locally uniformly to the Riemann zeta function in the critical strip. It approximates the trivial and non-trivial zeros of the Riemann zeta function. Some properties of the generalized Riemann zeta function are investigated. The relation between the function and the general Hurwitz zeta function is exploited to deduce new identities. Keywords Riemann zeta function Hurwitz zeta function Polylogarithm function Extended Fermi-Dirac Bose-Einstein 1 Introduction The family of zeta functions including Riemann Hurwitz Lerch and their generalizations constantly find new applications in different areas of mathematics number theory analysis numerical methods etc. and physics quantum field theory string theory cosmology etc. . A useful generalization of the family is expected to have wide applications in these areas as well. Some extensions of the Fermi-Dirac FD and Bose-Einstein BE functions have been introduced in 1 . The extended Fermi-Dirac eFD X 1 e-vt v s x t - x s - s-dt s 0 x 0 v -1 r s e 1 x and the extended Bose-Einstein eBE functions X 1 ẽ vv p s x t - x s-1 dt A J r s e - 1 x i v -1 i s 1 when x 0 -A s 0 when x 0 provide a unified approach to the study of the zeta family. These functions proved useful in providing simple and elegant proofs of some known results and .
