Báo cáo toán học: "Combinatorial proof of a curious q-binomial coefficient identit"

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: Combinatorial proof of a curious q-binomial coefficient identity. | Combinatorial proof of a curious q-binomial coefficient identity Victor J. W. Guoa and Jiang Zengb Department of Mathematics East China Normal University Shanghai 200062 People s Republic of China jwguo@ http jwguo bUniversite de Lyon Universite Lyon 1 Institut Camille Jordan UMR 5208 du CNRS 43 boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex France zeng@ http zeng Submitted Sep 18 2009 Accepted Feb 2 2010 Published Feb 8 2010 Mathematics Subject Classifications 05A17 05A30 Abstract Using the Algorithm Z developed by Zeilberger we give a combinatorial proof of the following q-binomial coefficient identity m-k M n k k m D-1 k 0 n Ẻ n k k 0 L -xq q k- 1 -mk 2 xm k- qmn 2 m k a a which was obtained by Hou and Zeng European J. Combin. 28 2007 214-227 . 1 Introduction Binomial coefficient identities continue to attract the interests of combinatorists and computer scientists. As shown in 7 p. 218 differentiating the simple identity V m . k V d x y m-k k t k k m x 7 k m x 7 n times with respect to y and then replacing k by m n k we immediately get the curious binomial coefficient identity V r J n kix -n- yk V e J n d x m-n-k x y k. mnk n mnk n mn n mn n k 0 v 7 v 7 k 0 v 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N13 1 Identity 1 has been rediscovered by several authors in the last years. Indeed Simons 13 reproved the following special case of 1 n fn I k n fn I kX nk n n k k n n k k 1 7 1 x k xk. 2 -J Is- M z -J Is- M rv rv rd rd k 0 v 7 v 7 k 0 v 7 v 7 Several different proofs of 2 were soon given by Hirschhorn 8 Chapman 4 Prodinger 11 and Wang and Sun 15 . As a key lemma in 14 Lemma Sun proved the following identity m r. I 7 n z I 7 _ 733 73 L Lr _ 73 733 L_ Lr -j m_k I m m I t n I k I z 1 I n k_a n Ị m I k m. k_a o I_1 vm 1 II I C Wt n v a I I I I zpm T a I Qi 1 7 1 1 x 7 x . 3 k M a I k M a I a a k 0 v 7 v 7 k 0 v 7 v 7 Finally by using the method of Prodinger 11 Munarini