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Báo cáo toán học: "Two finite forms of Watson’s quintuple product identity and matrix inversion"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Two finite forms of Watson’s quintuple product identity and matrix inversion. | Two finite forms of Watson s quintuple product identity and matrix inversion X. Ma Department of Mathematics SuZhou University SuZhou 215006 P.R.China Submitted Jan 24 2006 Accepted May 27 2006 Published Jun 12 2006 Mathematics Subject Classification 05A10 33D15 Abstract Recently Chen-Chu-Gu 4 and Guo-Zeng 6 found independently that Watson s quintuple product identity follows surprisingly from two basic algebraic identities called finite forms of Watson s quintuple product identity. The present paper shows that both identities are equivalent to two special cases of the Ợ-Chu-Vandermonde formula by using the -inversion. 1. Introduction The celebrated Watson s quintuple product identity 5 Exer. 5.7 p. 147 states that for q 1 and a variable z X z2q2k 1 - 1 z3k 1qk 3k 1 2 q z q z q i qz2 q z2 q2 1.1 k i where the q-shifted factorial a q n is dehned by n 1 a q n ỊỊ 1 - aqk 1 and a q i ỊỊ 1 - aqk k 1 k 0 with the following compact multi-parameter notation a1 a2 am q n a1 q n a2 q n am q n- This identity has several important applications in combinatorial analysis number theory and special functions. For its historical note and various proofs we refer the THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R52 1 reader to 2 7 . Recently as a new and perhaps the simplest proof Chen Chu and Gu 4 found that it can be derived simply from the following almost-trivial algebraic identity called a finite form of the quintuple product identity the latter is just a limiting case of the terminating q-Dixon formula 5 II-14 . Theorem 1 Finite form of Watson s quintuple product identity For any integer n 0 and a variable z there holds n E 1 zqk k 0 n z q n i zk F -. kq z2qk q n 1 1.2 where n q denotes the q-binomial coefficient n q q q n q q k q q n-k Not much later a somewhat different finite form of this identity was independently discovered by Guo and Zeng 6 Theorem 8.1 via recurrence approach. Their result is Theorem 2 Another finite form of Watson s quintuple product identity For n 0