tailieunhanh - Báo cáo toán học: "A note on an identity of Andrews"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A note on an identity of Andrews. | A note on an identity of Andrews Zhizheng Zhang Department of Mathematics Luoyang Teachers College Luoyang 471022 P. R. China zhzhzhang-yang@ Submitted Jan 26 2005 Accepted Feb 23 2005 Published Mar 7 2005 Mathematics Subject Classifications 33D15 05A30 Abstract In this note we use the -exponential operator technique on an identity of Andrews. 1 Introduction The following formula is equivalent to an identity of Andrews see 3 or 1 q bc acdf q n _ cX q bd acdf q n bc n ad df q n i c n 0 ac cf q n i d q qd c c d abcd acdf bcdf q ac ad cf df bc bd q 1 1 Liu 3 showed it can be derived from the Ramanjan YỘ1 summation formula by the q-exponential operator techniques. In this short note again using the q-exponential operator technique on it we obtain a generalization of this identity. We have Theorem . Let 0 1 q 1. Then d X q bc q ce acdf q n qn _ c X q bd q de acdf q n qn ad df q n i q2 bcde q n ac cf q n i q2 bcde q n n o n v d q qd c c d abcd acdf bcdf acde cdef bcde q q 1 ac ad cf df bc bd ce de abc2d2ef q q 1 2 This research is supported by the National Natural Science Foundation of China Grant No. 10471016 . THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N3 1 2 The proof of the Theorem The -difference operator and the q-shift operator q are defined by Dq f a 1 f a - f aq a and e f a f aq respectively. In 2 Chen and Liu construct the operator e E Dq Based on these they introduce a q-exponential operator E be X n 0 be n q n q q n For E be there hold the following operator identities. Applying we rewrite 1 as E be at q i at bt q i ỈE Ũ 1 _ as at bs bt q i E be as at q i 7--------------- abst q q i q a q n -a nq n 1 q na q i a q i dX acdf q n n 0 ad df q n i .nX acdf q n nn n 1 i i cA accf q i i cd q qq bd-bc q1 q qd c c d acdf q i ----- abcd bcdf q i ac ad cf df q i 3 4 5 6 n Applying E ee to both sides of the equation with respect to the variable b gives acdf q n ad df q n i - A q E ee q-nbc bd q i d -c n 0 acdf q n ac cf q n i -0nq n 1 E ee q-nbd .