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Báo cáo toán học: "A Note on The Rogers-Fine Identity"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Note on The Rogers-Fine Identity. | A Note on The Rogers-Fine Identity Jian-Ping Fang Department of Mathematics Huaiyin Teachers College Huaian Jiangsu 223300 P. R. China Department of Mathematics East China Normal University Shanghai 200062 P. R. China fjp7402@163.com Submitted May 29 2006 Accepted Jul 30 2007 Published Aug 9 2007 Mathematics Subject Classihcations 05A30 33D15 33D60 33D05 Abstract In this paper we derive an interesting identity from the Rogers-Fine identity by applying the q-exponential operator method. 1 Introduction and main result Following Gasper and Rahman 7 we write a q o 1 a q n 1 - a 1 - aq 1 - aqn 1 n 1 1 1 b1 ar x t a1 a2 ar q n bs q x A q b1 A q n 1 nqn n-1 2j 1 s r xn. For convenience we take q 1 in this paper. Recall that the Rogers-Fine identity 1 2 6 10 is expressed as follows X n q n _n V1 n q n qnr q n 1 - arq2n _ 2_ n 0H q nT n 0 H q n T q n 1 T q 1 This identity 1 is one of the fundamental formulas in the theory of the basic hypergeometric series. In this paper we derive an interesting identity from 1 by applying the q-exponential operator method. As application we give an extension of the terminating very-well-poised 6 5 summation formula. The main result of this paper is Jian-Ping Fang supported by Doctorial Program of ME of China 20060269011. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N17 1 Theorem 1.1. Let a_1 a0 a1 a2 a2t 2 be complex numbers a2i I 1 with i 0 1 2 t 1 then for any non-negative integer M we have yX q M c a2 a4 a2t 2 q n Tn n 0Ua1 3 a2t i q n XX q M q m q1-M P q m 1 - 2 . m 0 H q m T q m 1 x TT1 a2j q m y q m q1 m p b c q mi y 0 a q m m 0 q q1-Mt A q1-m c q mi 0 mi 2 m . m2 mi I 1 q q1 m a2i-3 a2i-1 a q q . 2 i i q q1-m a2i q1 m a2i-2 a2i-3 qLt 1 2 where t 1 0 1 2 1 c a0 and b a_1. 2 The proof of the theorem and its application Before our proof let s first make some preparations. The q-differential operator Dq and q-shifted operator q see 3 4 8 9 acting on the variable a are defined by Dq f a g f a f aq and q f a g f aq a Rogers 9 first .