# Chapter 16 of Ramanujan's second notebook: theta-functions and q-series

Chandrashekara Adiga, and Berndt, Bruce C. and Bhargava, S. and Watson, G. N. (1985) Chapter 16 of Ramanujan's second notebook: theta-functions and q-series. Mem. Amer. Math. Soc., 53 (315). v+85.

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## Abstract

Almost everything Ramanujan looked at was changed by some of his work. Some topics were only changed slightly, while in other areas Ramanujan completely revised the way we look at it. One place where he made major changes was q-series, or more technically, basic hypergeometric series. These are series Σcn with cn+1/cn a rational function of qn for fixed q with |q|<1. Most of Ramanujan's deepest work on these series was done late in his life, and is in the work Andrews found and calls the "lost notebook''. The present chapter is the first one in the published notebooks to deal with q-series. The first half is a summary of much of the elementary theory of basic hypergeometric series, including some series transformations, continued fractions (which come from three-term recurrence relations), and explicit evaluation of sums up to the level of the very well poised 0φ5. One of the series transformations is a double limit of Watson's extension of Whipple's transformation, and as is relatively well known, this implies the Rogers-Ramanujan identities. Thus Ramanujan knew enough to prove these identities although he was unaware of it. One particularly important identity is Ramanujan's bilateral extension of the q-binomial theorem, his 1ψ1 sum. This is a q-extension of the beta integral on [0,∞], just as the q-binomial series is a q-extension of the beta integral on [0,1]. There is a very illuminating proof of the 1ψ1 identity in the present paper, which was found by Berndt. Watson's proof was different, and is probably the proof that Hardy referred to. The second half of this chapter deals with theta functions, and it has to be read to be appreciated. Using nothing more than mathematics known to freshman calculus students and the 1ψ1 sum, Ramanujan developed many of the classical identities of theta functions in one variable, and a number of completely new results. A couple of the most striking new results are ones he sent in his first letter to Hardy, and about which Hardy said he had never seen anything like them, but that they must be true for no one could have the imagination to think up something like them. One of these is e−2π/51+e−2π1+e−4π1+⋯=5+5√2−−−−−−−√−5√+12 where the left-hand side is one of the standard notations for a continued fraction. If one is interested in the question of what the human mind can do, it pays to look seriously at Ramanujan's accomplishments, for his life and work are a singular example of what can be done despite adversity. The published notebooks have stood as a vast array of work, but without work such as Berndt has done in this and previous papers, its place in the rest of mathematics could not be determined. One suspected there were gems that had not been discovered there (and there were), and one hoped that some idea of what made Ramanujan tick might arise. We will probably never learn this in any detail, but there are vague glimmerings every once in a while that come through better in the notebooks than they do in the published papers. Berndt deserves our thanks for his labor in this project, and the Vaughn Foundation has shown good sense in helping to support this work.

Item Type: Article Physical Sciences > Mathematics PG Campuses > Manasagangotri, Mysore > Mathematics Kodandarama 21 May 2013 03:07 26 Oct 2015 05:59 http://eprints.uni-mysore.ac.in/id/eprint/9771