When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the

The paper classifies algebraic transformations of Gauss hypergeometric functions and pull-back transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.

. There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function 2 F 1 ( 1 3 ; 2 3 1 ; \Delta) analogous to the classical ` 2 (q), ` 3 (q), ` 4 (q) and the hypergeometric function 2 F 1 ( 1 2 ; 1 2 1 ; \Delta). We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function `(z; q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity. The second author completed this research while an NSERC International Fellow at Dalhousie. The third author was supported by NSERC of Canada. Received by the editors December 13, 1991. AMS Classification: 33D10. Secondary: 05A19, 11B65, 11F27, 33C05 c flCanadian Mathematical Society 1993. 1 x 1 Introduction and statement of results In a recent paper, Borwein, Borwein and Garvan [B-B-G] introduce three functions, a(q) = X q m 2 +mn+n 2 ; b(q) = X ! m\Gamman q m 2 +m...

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1. Introduction. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper [BBG] in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by Ramanujan in his unpublished notebooks. No record of Ramanujan’s original

this paper is to prove each of these new evaluations. We also establish some new evaluations of '(q)

. In his lost notebook, Ramanujan recorded certain identities involving integrals of theta functions. Some identities are proved by using identities relating theta functions and Lambert series. Two identities give integral representations for the Rogers-Ramanujan continued fraction. 1. Introduction On pages 207 and 46 in his lost notebook [9], Ramanujan recorded eight identities involving integrals of theta functions. Two of these give integral representations for the Rogers-Ramanujan continued fraction, defined by F (q) := 1 1 + q 1 + q 2 1 + q 3 1 + \Delta \Delta \Delta jqj ! 1: More precisely, on page 46, Ramanujan claims that q 1=5 F (q) = p 5 \Gamma 1 2 exp ` \Gamma 1 5 Z 1 q (1 \Gamma t) 5 (1 \Gamma t 2 ) 5 \Delta \Delta \Delta (1 \Gamma t 5 )(1 \Gamma t 10 ) \Delta \Delta \Delta dt t ' = p 5 \Gamma 1 2 \Gamma p 5 1 + 3 + p 5 2 exp ` 1 p 5 Z q 0 (1 \Gamma t) 5 (1 \Gamma t 2 ) 5 \Delta \Delta \Delta (1 \Gamma t 1=5 )(1 \Gamma t ...

Beginning with Ramanujan’s epic paper, Modular Equations and Approximations to π, we describe Ramanujan’s series for 1/π and later attempts to prove them. Generalizations, analogues, and consequences of Ramanujan’s series are discussed.

New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1

In his lost notebook, Ramanujan defined a parameter λn by a certain quotient of Dedekind eta-functions at the argument q = exp(−π √ n/3). He then recorded a table of several values of λn. To prove these values (and others), we develop several methods, which include modular equations, the modular j-invariant, Kronecker’s limit formula, Ramanujan’s “cubic theory ” of elliptic functions, and an empirical process. 1. Introduction. On the top of the page 212 in his lost notebook, Ramanujan defined the function λn by (1.1)

Abstract. Using certain representations for Eisenstein series, we derive several of Ramanujan’s series for 1/π arising from his cubic and quartic theories of elliptic functions.