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Báo cáo toán học: "RAMANUJAN’S METHOD IN q-SERIES CONGRUENCES"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: học" RAMANUJAN’S METHOD IN q-SERIES CONGRUENCES. | RAMANUJAN S METHOD IN Ợ-SERIES CONGRUENCES By George E. Andrews 1 and RANjAN Roy Written in honor of Herb Wilf s 65th birthday Abstract. We show that the method developed by Ramanujan to prove 5 p 5n 4 and 7lp 7n 5 may in fact be extended to a wide variety of ợ-series and products including some with free parameters. 1. Introduction. Ramanujan 11 is the discoverer of the surprising fact that the partition function p n satishes numerous congruences. Among the inhnite family of such congruences the two simplest examples are 1.1 p 5n 4 0 mod 5 and 1.2 p 7n 5 0 mod 7 . Ramanujan used an ingenious and elementary argument to prove these congruences which relied on Jacobi s famous formula 10 last eqn. p.5 1.3 q q 1 n 1 - qn 3 X -1j 2j 1 qjj 1 2 n 1 j 0 where N-1 Ơ-4 a n A q N n 1 - Aqj . j 0 1 Partially supported by National Science Foundation Grant DMS-8702695-04 Typeset by Ạ S-TgX A rather more general result of this nature was proved in 3 p. 27 Th. 10.1 to account for certain congruences connected with generalized Frobenius partitions. Indeed J. M. Gandhi 7 8 9 J. Ewell 5 L. Winquist 12 and many others cf. Gupta 10 Sec. 6.3 have proved partition function congruences based on this idea. In all these theorems the underlying generating functions were either modular forms or simple linear combinations thereof. The point of this paper is to show that Ramanujan s original method is applicable to an inhnite number of congruence theorems including many non-modular functions dehned by q-series. Our main result is Theorem 1. Suppose p is a prime 3 and 0 a p and b are integers. Also a must be a quadratic nonresidue mod p. Suppose on 1 1 an z1 z2 . Zj is a doubly infinite sequence of Laurent polynomials overT with variables Z1 . Zj independent of q. Then there is an integer c such that the coefficient of zm1 zm2 zjm qpN in qc Ể qa n bn 1 5 q is divisible by p. For each integer m we shall denote by m the multiplicative inverse of m mod p. The integer c cp a b may be chosen as the .