this paper, from which we can derive two formulas for the Rogers-Ramanujan continued fraction found on page 208 in Ramanujan's lost notebook. They are the first known formulas for computing directly the values of the Rogers-Ramanujan continued fraction. The proofs of these formulas are given in Section 3. ROGERS-RAMANUJAN CONTINUED FRACTION AND THETA-FUNCTIONS 3

this paper is to establish in Section 2 each of these eighteen values, and the tools we shall use were entirely known to Ramanujan. The methods we develop can be easily applied to determine further values of a m;n ; and we offer some of these. Upon examining the table, we see that when (m; n) = 1 each value is a unit in some algebraic number field. Thus, our second goal in Section 3 is to prove that a m;n is a unit for large classes of pairs m; n: We also establish in Section 3 some general formulas for evaluating a m;n : Here, we use ideal class theory, with which Ramanujan was likely unfamiliar. After stating his last value for a m;n on page 339, Ramanujan offers some values for '(e

Introduction As usual, set (a; q) 1 = 1 Y n=0 (1 \Gamma aq n ); jqj ! 1; and, following Ramanujan, let (q) = (\Gammaq; q 2 ) 1 : If n is any postitive rational number and q = exp(\Gamma p n); the two class invariants G n and g n are defined by (1.1) G n := 2 \Gamma1=4 q \Gamma1=24 (q) and g n := 2 \Gamma1=4 q \Gam

this paper. We also do not separately list in our references the solvers of the problems cited in Ramanujan's Collected Papers [172]. However, if a solution was published after the publication of the Collected Papers in 1927, then we record it as a separate item in the bibliography. Many of the problems, or portions thereof, can be found in Ramanujan's notebooks [171]. Normally in such a case, we cite where a problem can be located in the notebooks and where it can also be found in Berndt's accounts of the notebooks [20]--[24].

Abstract. Many of Ramanujan’s ideas and theorems form the seeds of questions and problems, many of which remain unresolved or even to be thoroughly examined. This survey raises questions arising from Ramanujan’s work on theta-functions and other q-series, with Gaussian hypergeometric functions making frequent appearances.