文档介绍：April 17, 2006 18:1 WSPC/INSTRUCTION FILE RRSijnt
International Journal of Number Theory
c World Scientific pany
IDENTITIES OF THE ROGERS-RAMANUJAN-SLATER TYPE
ANDREW V. SILLS
Department of Mathematics, Rutgers University, Hill Center, Busch Campus
Piscataway, New Jersey 08854-8019, USA
******@
Received (19 August 2005)
Revised (17 April 2006)
Accepted (Day Month Year)
Communicated by Bruce C. Berndt
It is shown that (two-variable generalizations of) more than half of Slater’s list of 130
Rogers-Ramanujan identities (L. J. Slater, Further identities of the Rogers-Ramanujan
type, Proc. London Math Soc. (2) 54 (1952), 147–167) can be easily derived using
just three multiparameter Bailey pairs and their associated q-difference equations. As a
bonus, new Rogers-Ramanujan type identities are found along with binatorial
interpretations for many of these identities.
Keywords: Rogers-Ramanujan identities; integer partitions; q-difference equations.
Mathematics Subject Classification 2000: 11P81, 11B65, 05A19, 39A13
1. Introduction
As usual, let us define
n−1
(A; q) := (1 − Aqj),
n Y
j=0
∞
j
(A; q)∞:= (1 − Aq ),
Y
j=0
(A1, A2, . . . , Ar; q)∞:= (A1; q)∞(A2; q)∞· · · (Ar; q)∞,
and assume throughout that |q| < 1.
The pair of q-series identities
∞ n2 2 2 4 4
q (q , q , q ; q )∞
= ()
X (q2; q2) (q; q)∞
n=0 n
and
∞ n(n+1) 3 4 4
q (q, q , q ; q )∞
= ()
X (q2; q2) (q; q)∞
n=0 n
1
April 17, 2006 18:1 WSPC/INSTRUCTION FILE RRSijnt
2 A. V. SILLS
are special cases of one of the earliest q-series identities, an identity due to Euler ([17,
Chapter 16], [4, p. 19, Eq. ()]):
∞ n
xnq(2)
= (−x; q)∞. ()
X (q; q)
n=0 n
One reason that identities () and () are significant is that they foreshadow a
pair of deeper identities discovered first by L. J. Rogers [27], and later independently
rediscovered by S. Ramanujan [26, Vol. 2, p. 33 ff.]:
Theorem (Th