Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Determinant Identity that Implies Rogers-Ramanujan. | A Determinant Identity that Implies Rogers-Ramanujan Kristina C. Garrett Department of Mathematics and Computer Science Carleton College Minnesota USA kgarrett@ Submitted Oct 2 2004 Accepted Nov 23 2004 Published Jul 29 2005 MR Subject Classifications 05A30 33C45 Abstract We give a combinatorial proof of a general determinant identity for associated polynomials. This determinant identity Theorem gives rise to new polynomial generalizations of known Rogers-Ramanujan type identities. Several examples of new Rogers-Ramanujan type identities are given. 1 Introduction The Rogers-Ramanujan identities are well known in the theory of partitions. They may be stated analytically as X qn n 0 1 n X 1n2 n s 7T - . q5 1 ------1 . . 1 2 where a q n 1 - a 1 - aq 1 - aqn 1 for n 0. a 1 0 1. and a 1 1 n 1 - a1n . n 0 1 1 a q i b 1 1. These identities were first proved by Rogers in 1894 13 Ramanujan and Rogers in 1919 14 and independently by Schur in 1917 15 . In particular Schur gave an ingenious proof that relied on the integer partition interpretation and used a clever sign-reversing THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R35 1 involution on pairs of partitions to establish the identities. Throughout the last century many proofs and generalizations have been given in the literature. For a survey of proofs before 1989 see 1 . In 5 we gave a generalization of the classical Rogers-Ramanujan identities writing the infinite sum as a linear combination of the infinite products in 1 and 2 . Theorem . For m 0 an integer .n2 mn -1 mq m Cm q _ -1 mq m dm q q5 i q5 i where Cm q dm q J2 -1 AqA 5A 3 2 J2 -1 AqA 5A 1 2 m-1 m 1 5A J . m-1 m 1 5A J 3 4 5 yq o q q n As usual xj denotes the greatest integer function and the q-binomial coefficients are defined as follows n m n L -J q qn 1 q m q q m 0 if m 0 is an integer otherwise 6 Í It is customary to omit the subscript in the case where it is q. In future use we will only include the subscript
