tailieunhanh - Báo cáo toán học: " Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type. | Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type Peter Paule Research Institute for Symbolic Computation Johannes Kepler University Linz A-4040 Linz Austria Submitted July 15 1994 Accepted July 26 1994. Abstract New short and easy computer proofs of finite versions of the Rogers-Ramanujan identities and of similar type are given. These include a very short proof of the first Rogers-Ramanujan identity that was missed by computers and a new proof of the well-known quintuple product identity by creative telescoping. AMS Subject Classification. 05A19 secondary 11B65 05A17 1 Introduction The celebrated Rogers-Ramanujan identities stated as series-product identities are k2 ak 1 y 7- q - Aị 1 - q 1 - q2 . 1 - qk 1 1 - q5j 1 1 - q5j- 4 1 where a 0 or a 1 see . Andrews 6 which also contains a brief historical account. It is well-known that number theoretic identities like these or of similar type can be deduced as limiting cases of q-hypergeometric finite-sum identities. Due to recent algorithmic breakthroughs see for instance Zeilberger 24 or Wilf and Zeilberger 23 proving these finite versions becomes more and more routine work that can be left to the computer. For instance the computers Ekhad and Tre 12 delivered a purely verification proof of the following finite version which was stated first in this form by Andrews 3 see also Bressoud 10 . X qfc2 X -1 kq 5k2-fc 2 2 V q q k q q n-k k q q n-k q q n k where a q n 1 - a 1 - aq 1 - aqn 1 for n 1 a q 0 1 and 1 q q n 0 for n 0. In the limit n 1 one gets identity 1 with a 0 after applying Jacobi s triple product identity see section . It is evident that the hard part of deducing for instance the Rogers-Ramanujan identities as a limiting case of a finite version consists in finding such a representation. On the other hand it is to expect that automated proving will enlarge the heuristic toolbox in a significant way. Nevertheless in this .