THE RAMANUJAN JOURNAL, Vol. 3, No. 1 (1999), 91-112.

Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan's lost notebook

SOON-YI KANG
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801
kang@math.uiuc.edu

Received June 5, 1997; Accepted September 2, 1998

Abstract. In his lost notebook, Ramanujan recorded several modular equations of degree $5$ related to the Rogers-Ramanujan continued fraction $R(q)$. We prove several of these identities and give factorizations of some of them in this paper. The parameter $k = R(q) R^2(q^2)$ introduced by Ramanujan in his second notebook has not been recognized for its usefulness. In this work we demonstrate how beautifully the parameter $k$ works, as we prove several identities stated by Ramanujan in the lost notebook.

Keywords: Rogers-Ramanujan continued fraction, modular equation, theta function, Ramanujan's lost notebook

1991 Mathematics Classification: Primary 33D10; Secondary 11F27