Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Binomial Coefficient Identity Associated with Beukers’ Conjecture on Ap´ry numbers e. | A Binomial Coefficient Identity Associated with Benkers Conjecture on Apery numbers CHU Wenchang College of Advanced Science and Technology Dalian University of Technology Dalian 116024 P. R. China Submitted Oct 2 2004 Accepted Nov 4 2004 Published Nov 22 2004 Mathematics Subject Classifications 05A19 11P83 Abstract By means of partial fraction decomposition an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity which in turn has been shown to imply Beukers conjecture on the congruence of Apery numbers. Throughout this work we shall use the following standard notation Harmonic numbers Shifted factorials II 0 x o 1 and and Hn n 1 1 k i x nn-1 x k I x n 1 k 0 x k for n 1 2 . For a natural number n let A n be Apery number defined by binomial sum A n X n n p 2 k 0 and a n determined by the formal power series expansion X rn qm q f 1 - q2n 4 1 - q4n 4 q - 4q3 - 2q5 24q7 . m 1 n 1 Beukers conjecture 3 asserts that if p is an odd prime then there holds the following congruence cf. 1 Theorem a p 2 a p mod p2 . The work carried out during the summer visit to Dalian University of Technology 2004 . THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N15 1 Recently Ahlgren and Ono 1 have shown that this conjecture is implied by the following beautiful binomial identity 2 X ỳ i k k 1 2kHn k 2kHn-k 4kHk 0 k 1 1 which has been conhrmed successfully by the WZ method in 2 . The purpose of this note is to present a new and classical proof of this binomialharmonic number identity which will be accomplished by the following general algebraic identity. Theorem. Let x be an indeterminate and n a natural number. There holds x i x n x n i 1 X n 2 n x k i k k -2kll. 2kHn-k-4kHk x k 2 The binomial-harmonic number identity 1 is the limiting case of this theorem. In fact multiplying by x across equation 2 and then letting x to we recover immediately identity 1 . Proof of the Theorem. By means of the standard
