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
