Báo cáo toán học: "The q-Binomial Theorem and two Symmetric q-Identities"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The q-Binomial Theorem and two Symmetric q-Identities | The q-Binomial Theorem and two Symmetric q -Identities Victor J. W. Guo Center for Combinatorics LPMC Nankai University Tianjin 300071 People s Republic of China jwguo@ Submitted Jun 30 2003 Accepted Sep 4 2003 Published Sep 12 2003 MR Subject Classifications 05A19 05A17 Abstract We notice two symmetric -identities which are special cases of the transformations of 2 1 series in Gasper and Rahman s book Basic Hypergeometric Series Cambridge University Press 1990 p. 241 . In this paper we give combinatorial proofs of these two identities and the ợ-binomial theorem by using conjugation of 2-modular diagrams. 1 Introduction We follow the notation and terminology in 7 and we always assume that 0 q 1. The q-shifted factorial is defined by n 1 a q o 1 a q n JỊ 1 - aqk n N a q ỊỊ 1 - aqk k 0 k 0 The following theorem is usually called the q-binomial theorem. It was found by Rothe and was rediscovered by Cauchy see 1 p. 5 . Theorem If z 1 then tt X a q n zn az q q q n z qK Various proofs are known. For simple proofs of see Andrews 3 Section and Gasper 6 and for combinatorial proofs see Alladi 2 and Pak 8 . The following two theorems are special cases of the transformations of 2Ộ1 series in Gasper and Rahman 7 p. 241 . THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R34 1 Theorem For a 1 and b 1 we have X az q n bn ho a q n 1 Theorem We have X q z q k z q n-kqmkzk k 0 q q k q q n-k q _ X bz q n n k b q n i XX q z q k z q m-kqnkzk k 0 q q k q q m-k q Clearly the left-hand side of may be written as 1 X 201 az q qa q b . 1 - a By the Heine s transformation in Gasper and Rahman 7 p. 241 is equal to 1 q abz q __1 2C1 a b abz q q 1 - a qa b q which is symmetric in a and b. Note that the special case z 0 of has also appeared in the literature see Stockhofe 9 and Pak 8 . Rewrite the left-hand side of as 7Z7q n 2h q-n q z q1-n z q qm 1 . q q n Applying the transformation in 7 p. 241 we get qmn 3Ộ2 .