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Báo cáo toán học: " Two Simple Proofs of Winquist’s Identity"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Two Simple Proofs of Winquist’s Identity. | Two Simple Proofs of Winquist s Identity Chutchai Nupet Department of Mathematics Faculty of Science Prince of Songkla University Hatyai Songkhla 90112 Thailand por075@hotmail.com Sarachai Kongsiriwong Department of Mathematics Faculty of Science Prince of Songkla University Hatyai Songkhla 90112 Thailand sarachai.k@psu.ac.th Submitted May 31 2010 Accepted Aug 12 2010 Published Aug 24 2010 Mathematics Subject Classification 11F20 11F27 Abstract We give two new proofs of Winquist s identity. In the first proof we use basic properties of cube roots of unity and the Jacobi triple product identity. The latter does not use the Jacobi triple product identity. 1 Introduction Winquist s identity was discovered by L. Winquist 6 in 1969. He used it to prove the congruence p 11n 6 0 mod 11 where p n is the number of partitions of the positive integer n. In 1972 L. Carlitz and M. V. Subbarao 1 gave a simple proof and a generalization of Winquist s identity. In 1987 M. D. Hirschhorn 3 gave another generalization of Winquist s identity. In 1997 S.-Y. Kang 4 gave a simple proof using the Jacobi triple product identity the quintuple product identity and two other identities from Ramanujan s notebooks. In 2003 S. Kongsiriwong and Z.-G. Liu 5 gave a simple proof using the Jacobi triple product identity and some properties of cube roots of unity. In this paper we give two new proofs of Winquist s identity for any complex number q with q 1 and any nonzero complex numbers a THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R116 1 and b E E 1 m nq 3m2 3n2 3m n 2 m - n - a- 3 b-3n a-3mb3n 1 a-3n 1b-3m-1 a3n 2b-3m-1 q q 2- a q a 1q q b q - b 1q q - ab q - a-1b-1 q q - ab-1 q - a-1bq q - 1-1 where a q denotes JJ 1 aqn-1 . In both proofs we will use the fact about u n 1 exp 2ni 3 that for any complex number a 1 a 1 au 1 au2 1 a3. 1.2 2 First Proof In this section we prove Winquist s identity by using some properties of cube roots of unity and the Jacobi triple product identity for each pair .