Resonance of Ramanujan’s Mathematics.

R. P. Agarwal. New Age International Publishers, 4835/24, Ansari Road, Daryaganj, New Delhi 110 002, India. 1999. Volume III. 216 pp. Price: Rs 400.

Ramanujan recorded his mathematical discoveries during 1903–1914 (prior to leaving for England) in his Notebooks. Plans to edit and publish the Notebooks simultaneously with the Collected
Papers of Srinivasa Ramanujan
(Cambridge, 1927) fell through for various reasons including financial constraints. In 1929, B. M. Wilson and G. N. Watson commenced editing the Notebooks, but after several chapters therein got edited, Wilson’s premature death in 1935 brought the mission to an abrupt standstill. Assiduous study of the Entries in the Notebooks (jinxed or not!) by several mathematicians continued unabated even as of Ramanujan’s
Papers
. Hardy’s detailed analysis of
Ramanujan’s Entries involving hypergeometric series triggered a veritable ‘flood of papers by Bailey, Watson, Whipple and others’. Mathematicians at large could experience the thrill of seeing the Notebooks ‘live’, with the publication in 1957 of a facsimile edition in 2 bound volumes, by the Tata Institute of Fundamental Research; volume 2 shall be referred to as RNB hereinafter.

The discovery of Ramanujan’s Lost Notebook in 1976 and Andrews’ own outstanding work on Ramanujan’s formulae therein on q-series and other topics greatly intensified study of
Ramanujan’s results; in the sequel, RLN shall refer to the Narosa publication (1988): Lost Notebook and Other Unpublished Papers by Srinivasa Ramanujan. Undaunted by the stupendous problem of editing the Notebooks, Berndt has successfully authored five beautiful volumes entitled Ramanujan’s Notebooks, unmatched for reader-friendly access to Ramanujan’s Notebooks.

Continued fractions, perhaps no hot favourites with mathematicians, had evidently fascinated Ramanujan immensely. (Incidentally, the ‘regular’ continued fraction expansion for ‘pi’ seems still unknown!) Hardy acknowledged having been ‘defeated completely’ by three of Ramanujan’s

continued fraction representations in Ramanujan’s first letter to Hardy in 1913. Chapter XII of RNB is mostly devoted to deep and beautiful formulae involving weird continued fractions; its Entry 34 and the following special case of its Entry 25 (due originally to Euler and Stieltjes), namely (where the right hand side represents an infinite continued fraction in standard notation and G on the left denotes Euler’s gamma function) did figure in Ramanujan’s aforesaid letter. A formidable Entry 40 therein involving a 5-parameter continued fraction is ‘one of Ramanujan’s crowning achievements in... continued fractions’. Watson gave its first published proof in 1935. According to Ramanathan (Proc. Indian Acad. Sci., 1987, 97, 277–296), ‘Ramanujan had not only rediscovered many of the results of Gauss, Euler, Heine and others on hypergeometric series and continued fractions but also found many new ones using systematically the three-term relations’. Also, ‘Ramanujan had thought of q-generalizations of (many of the classical) results...’, as his notings on ‘left hand side pages’ in (vol. 1 of) the (TIFR) Notebooks indicate.

The first three chapters of the book under review contain the Entries dealing with continued fractions and q-extensions in Chapters XII, XVI and the ‘unorganized portion’ of RNB and in RLN meticulously picked by the author and supplemented with expert comments and proofs/outlines of proof. His penchant for proofs by ‘hypergeometric methods’ and intensive application of ‘contiguous relations’ among ordinary or basic hypergeometric series comes alive all through. Gauss’ ‘contiguous’ relation

 

F (a, b; c; z) = F (a, b + 1; c; z)

– (az/c)F (a + 1, b+ 1; c + 1; z)

 

is a simple three-term relation linking Gauss’ hypergeometric series F in 3 parameters a, b, c with two others

‘contiguous’ to F. Evidently, such ‘contiguous’ relations are handy tools to obtain Gauss’ continued fraction representation for the quotient F(ab + 1; c + 1; z)/F(abcz). But not so obvious is that ‘contiguous’ relations help in proving the formula (*) cited above; Ramanathan’s ‘hypergeometric’ proof for Entry 25 of Chapter XII of RNB uses such relations via Euler’s continued fraction representation for quotients of hypergeometric series, Kummer summation, etc. The author is justified, of course, in preferring the deduction of Entry 25 from a ‘hypergeometric proof’ for Entry 33 of Chapter XII by Denis and Singh applying ‘contiguous’ relations among (5-parameter) 3F2(1)’s. He also lauds ‘uniform’ proofs based on Wilson’s ‘contiguous’ relations for 3F2(1)’s in respect of 3 other Entries in Chapter XII and the ‘unifying methods’ of Bhargava, Denis and others (involving quotients of basic hypergeometric series) which yield, as special cases, Entries in Chapter XVI (of RNB) such as on the Rogers–Ramanujan continued fraction besides results of Gordon, Hirschhorn, etc. Ramanujan’s formulae involving Rogers–Ramanujan type continued fractions (with 2 parameters ab), their specific ‘singular values’, modular equations are also found here; the case a = b = 0 corresponds to the celebrated Rogers–Ramanujan(-Schur) continued fraction and precisely the same 9 cases of these 2-parameter continued fractions were identified by A. Selberg independently in 1936 as ones admitting infinite product expansions. The Entries cited in lines 9–10 of page 93 refer, by the way, to one and the same formula of Ramanujan’s!

Entries in RNB concerning Riemann’s zeta function z are covered by Chapter 4 in the context of continued fractions for zeta values, asymptotic expansions, etc. The Entry giving the functional equation of z could have been highlighted as a thriller and ‘Ramanujan’s famous formula’ for z (2n + 1) also accommodated. Though Bhargava’s interesting article reproduced in Chapter 5 deserves careful treatment, we briefly mention that it provides an approach to Jacobian elliptic functions, product expansions and modular forms via two basic formulae of Ramanujan’s, cubic analogues of theta functions (cf. Patterson’s analogue in Crelle 1977), and elliptic functions to alternative signatures. Pages 189, 190 and 196 contain minor misprints!

Finally, our wish list would include an Index to Terminology/Notation, indication of relevant page numbers for formulae recalled or referred to and a single Bibliography saving one from the mortification of finding the same paper being cited differently (e.g. Watson’s 1935 paper as Watson [2] on page 88 and Watson [1] elsewhere!).

Misprints are unavoidable but we point out a few: the variable z missing from (2), (2a), (3), (3a) on page 27 and (15) on page 29 and likewise a factor c missing from (14) on page 29. Minor corrections seem necessary for (49) on page 20, (11) on page 29, (16) on page 30 and Entry 11 on page 110. The definition for y on page 26 needs correction and may then be advanced to page 14 at its first occurrence and a uniform font for this y function be used throughout. One can understand Apéry being spelt wrongly but how does one stand Riemann being consistently misspelt in the Contents?

S. RAGHAVAN

A1, ‘Ashok Ganapathim’,

25, Fourth Main Road,

Raja Annamalaipuram,

Chennai 600 028, India