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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujanâ€™s tau-function Ï„(n).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 theâ€¦ (More)

- Bruce C. Berndt, Frank G. Garvan, Frank G. Garvan, Ã¼-.fiÃŸ
- 1995

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 theseâ€¦ (More)

- Bruce C. Berndt, Alexandru Zaharescu, âˆš Neâˆ’log
- 2002

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 oneâ€¦ (More)

- Bruce C. Berndt
- 2002

- George E. Andrews, Bruce C. Berndt, JAEBUM SOHN
- 2001

The research described in this paper was motivated by an enigmatic entry in Ramanujanâ€™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 âˆž in the three residue classes modulo 3, the nth partial quotients tend, respectively,â€¦ (More)

- Bruce C. Berndt
- 2002

This evaluation can be found in standard tables of series, such as those of E. R. Hansen [44, p. 262, eq. (30.1.2)], L. B. W. Jolly [56, pp. 102â€“103, eq. (352)], and A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev [70, p. 646, eq. 6]. We do not know who first proved (1.1), but several proofs exist. If we replace the power 2 on the left side by anâ€¦ (More)

- Nayandeep Deka Baruah, Bruce C. Berndt
- Journal of Approximation Theory
- 2009

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/ 2. Â© 2008 Elsevier Inc. All rights reserved. MSC: 33C05; 33E05; 11F11; 11R29

Many of Ramanujanâ€™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 Ramanujanâ€™s work on theta-functions and other q-series, with Gaussian hypergeometric functions making frequent appearances.

- Bruce C. Berndt, Heng Huat Chan, HENG HUAT
- 1999

A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujanâ€™s assertions about tn by establishing new connections between the modular jâˆ’invariant and Ramanujanâ€™s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbertâ€¦ (More)

My guess is that, within fifty or hundred years (or it might be one hundred and fifty) computers will successfully compete with the human brain in doing mathematics, and that their mathematical style will be rather different from ours. Fairly long computational verifications (numerical or combinatorical) will not bother them at all, and this should lead notâ€¦ (More)