Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A BINOMIAL COEFFICIENT IDENTITY ASSOCIATED TO A CONJECTURE OF BEUKERS. | THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R10 A BINOMIAL COEFFICIENT IDENTITY ASSOCIATED TO A CONJECTURE OF BEUKERS Scott Ahlgren Shalosh B. Ekhad Ken Ono and Doron Zeilberger Abstract. Using the WZ method a binomial coefficient identity is proved. This identity is noteworthy since its truth is known to imply a conjecture of Beukers. Received January 28 1998 Accepted February 1 1998 If n is a positive integer then let A n X a k 0 k k 2 and define integers a n by XX a n qn q ỊỊ 1 - q2n 4 1 - q4n 4 q - 4q3 - 2q5 24q7 - 1 Beukers conjectured that if p is an odd prime then 1 2 a p mod p2 . In A-O it is shown that 1 is implied by the truth of the following identity. Theorem. If n is a positive integer then n 2 2 n k n-k k ỷ kC U Y1 ỳ1 - 2 ỳ 1 0. fci k 2k ị Ị i 2 1 í f if Remark. This identity is easily verified using the WZ method in a generalized form Z that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n since it holds for n 1 2 3 check and since the sequence defined by the sum satisfies a certain homog. third order linear recurrence equation. To find the recurrence and its proof download the Maple package EKHAD and the Maple program zeilWZP from http zeilberg . Calling the quantity inside the braces c n k compute the WZ pair F G where F c n k 1 - c n k and G c n 1 k - c n k . Go into Maple and type read zeilWZP zeilWZP k n k 2 k 4 n-k 2 F G k n N References A-O S. Ahlgren and K. Ono A Gaussian hypergeometric series evaluation and Apery number congruences in preparation . B F. Beukers Another congruence for Apery numbers J. Number Th. 25 1987 201-210. Z D. Zeilberger Closed Form pun intended Contemporary Mathematics 143 1993 579-607. Department of Mathematics Penn State University University Park Pennsylvania 16802 E-mail address ahlgren@ Department of Mathematics Temple University Philadelphia Pennsylvania 19122 E-mail address ekhad@ http .
