Bruce C. Berndt

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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n).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 foregoing part of the(More)
In his famous paper on modular equations and approximations to n , Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories" in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recently proved by J. M. and P. B. Borwein in 1987, but these(More)
Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is pS(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then pS(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one(More)
The research described in this paper was motivated by an enigmatic entry in Ramanujan’s lost notebook [11, p. 45] in which he claimed, in an unorthodox fashion, that a certain q-continued fraction possesses three limit points. More precisely, he claimed that as n tends to ∞ in the three residue classes modulo 3, the nth partial quotients tend, respectively,(More)
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.
A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between the modular j−invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert(More)