tailieunhanh - Báo cáo hóa học: " Research Article An Interchangeable Theorem of q-Integral"

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: Research Article An Interchangeable Theorem of q-Integral | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 135693 8 pages doi 2009 135693 Research Article An Interchangeable Theorem of q-Integral Hongshun Ruan Department of Applied Mathematics Jiangsu Polytechnic University Changzhou City 213164 Jiangsu Province China Correspondence should be addressed to Hongshun Ruan rhs@ Received 25 August 2008 Accepted 3 January 2009 Recommended by Ondrej Dosly We give a sufficient condition for the interchangeability of the order of sum and q-integral by using inequality technique. As the application of the theorem some interesting results on the hypergeometric series are obtained. Copyright 2009 Hongshun Ruan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction and Some Lemmas q-series which are also called basic hypergeometric series plays a very important role in many fields such as affine root systems Lie algebras and groups number theory orthogonal polynomials and physics. Inequality technique is one of the useful tools in the study of special functions. There are many papers about it see 1-6 . First we recall some definitions notations and known results which will be used in this paper. Throughout this paper it is supposed that 0 q 1. The q-shifted factorials are defined as a q c 1 n-1 a q n n 1 - aqk n 1 2 . k 0 TO q n 1 - aqk h k 0 We also adopt the following compact notation for multiple q-shifted factorial a1 a2 . . . am q n a1 q n a2 q n amz q n where n is an integer or TO. 1-1 1-2 2 Journal of Inequalities and Applications The q-binomial theorem 2 tells us that ỳ a q kzk az q ả q q k z qL z 1. Replace a with 1 a and z with az and then set a 0 we get ỷ -1 kqk k-1 2zk à q qik z qK. Heine 2 introduced the basic hypergeometric series 2Ộ1 which is defined by 2 1 a1 a2 b1 q z aq k .