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Ramanujan’s lost notebook. Part II. (English) Zbl 1180.11001

New York, NY: Springer (ISBN 978-0-387-77765-8/hbk; 978-0-387-77766-5/ebook). xii, 418 p. (2009).
This book is the second of 4 projected volumes providing a critical edition (mainly providing proofs) of the entries in the “Lost Notebook”, discovered in the papers of G. N. Watson at Trinity College Library in Cambridge in 1976 by the first author. (See the Introduction to Part I, Zbl 1075.11001, for the material’s provenance.) As the Introduction to this makes clear – this was sotto voce in Part I’s Introduction – what is being redacted in these volumes is not merely the lost notebook, but also various fragments, letters and other unpublished material reproduced by Narosa in facsimile form in 1988 (ref. 244 of Part II).
As such this is a very heterogeneous collection of entries. The authors state in the Introduction that “Two primary themes permeate [herein]…, \(q\)-series and Eisenstein series.”
That said, it is not possible in a review of this length to summarize or even indicate the contents. There are 16 Chapters and even providing their titles (such as “12. Letters from Matlock House”, and, 7. Special Identities”) is not useful.
However, as I wrote this, the more useful entire list of 84 chapter sub-sections is viewable online using amazon.com’s “Search Inside” feature. (http://www.amazon.com/Ramanujans-Lost-Notebook-Part-Pt/dp/0387777652 Click Table at Contents.)
Finally, the reviewer is awestruck by the thoroughness of the authors. Here is but one example:
“Entry 16.1.1 \(\sum^\infty_{k=0}\frac {k^rq^{k^s}}{1-q^{k^s}}\)”
(Ramanujan actually wrote out the first three terms, then \(+\dots)\).
The book devotes a page ruminating on what this fragment might mean, finally but tentatively concluding it was a false start. Bravo!

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F11 Holomorphic modular forms of integral weight
11F27 Theta series; Weil representation; theta correspondences
11A55 Continued fractions
11P81 Elementary theory of partitions
33-02 Research exposition (monographs, survey articles) pertaining to special functions
33-03 History of special functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
05A17 Combinatorial aspects of partitions of integers
33E05 Elliptic functions and integrals
05A30 \(q\)-calculus and related topics

Citations:

Zbl 1075.11001
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