tailieunhanh - Báo cáo toán học: "Franklin’s argument proves an identity of Zagier"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Franklin’s argument proves an identity of Zagier. | Franklin s argument proves an identity of Zagier Robin Chapman School of Mathematical Sciences University of Exeter Exeter EX4 4QE UK rjc@ Submitted September 28 2000 Accepted November 9 2000 Abstract Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin s classical proof of Euler s pentagonal number theorem. Mathematics Subject Classification 2000 05A17 11P81 1 Introduction We use the standard q-series notation n a n n 1 - aqk-1 k 1 where n is a nonnegative integer or n 1. Euler s pentagonal number theorem states that q i 1 Ể -1 r q . 1 r 1 Recently Zagier proved the following remarkable identity Theorem 1 k E q i - q n q i E Vk E 1 r 3r - 1 qr 3r-1 2 3rqr 3r 1 2 . 2 n 0 k 1 q r 1 This is 8 Theorem 2 slightly rephrased. Equation 1 has a combinatorial interpretation. The coefficient of qN in q 1 equals de N do N where de N respectively do N is the number of partitions of N into an even respectively odd number of distinct parts. Franklin 4 showed that d N d N 1 r if N 2 r 3r 1 for a positive integer r o I 0 otherwise. THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R54 2 His proof was combinatorial. He set up what was almost an involution on the set of partitions of N into distinct parts. This involution reverses the parity of the number of parts. However there are certain partitions for which his map is not defined. These exceptional partitions occur precisely when N 1 r 3r 1 and so account for the nonzero terms on the right of 1 . Franklin s argument has appeared in numerous textbooks notably 1 and 5 . We show that Zagier s identity has a similar combinatorial interpretation which miraculously Franklin s argument proves at once. The author wishes to thank George Andrews and Don Zagier for supplying him with copies of 3 and 8 and also an anonymous referee for helpful comments. 2 Proof of Theorem 1 We begin by recalling Franklin s involution . Let DN denote the set of .