Ramanujan’s notebooks. Part I.

*(English)*Zbl 0555.10001
New York etc.: Springer-Verlag. X, 357 p. DM 188.00 (1985).

The book under review is Part I of a very important project undertaken by the author. In 1957, the Tata Institute of Fundamental Research in Bombay published unedited photostatic copies of Ramanujan’s Notebooks. These notebooks were prepared in the years 1903-1913 by Ramanujan before he went to England for his historic collaboration with G. H. Hardy. In the 1930’s, G. N. Watson and B. M. Wilson started to edit the notebooks; unfortunately Wilson died and the project was abandoned. Finally the mathematical community has available a readable version of Ramanujan’s Notebooks together with rigorous proofs (seldom even hinted at in the original notebooks) and relevant references. This is a beautiful job, and every mathematician owes Bruces Berndt warm thanks.

Chapter 1, Magic Squares, is probably representative of very early work and does not contain anything particularly surprising. Chapter 2, Sums related to the Harmonic Series or the Inverse Tangent Function, is also elementary; the formulas are still intriguing, e.g. \[ \sum^{\infty}_{k=0}\arctan (2/(n+2k+1)^ 2)= \arctan (1/n). \] Chapter 3, Combinatorial Analysis and Series Inversions, contains some very important results including alternatives to Lagrange inversion (see especially Entries 10 and 15); here we find that the author has gone to considerable trouble to provide rigorous corrected versions of Ramanujan’s results. In Chapter 4, Iterates of the Exponential Function and an Ingenious Formal Technique, we find the formal identity \[ \int^{\infty}_{0}x^{n-1}\quad \sum^{\infty}_{k=0}\phi (k) (- x)^ k dx=\pi \phi (-n)/\sin \pi n,\quad n>0; \] a rigorous discussion of this and related results is given. Chapter 5, Eulerian Polynomials and Numbers, Bernoulli Numbers, and the Riemann Zeta-Function, again provides a mathematically rigorous discussion of some interesting but unclear assertions by Ramanujan. The material in Chapter 6, Ramanujan’s Theory of Divergent Series, has been discussed previously by G. H. Hardy [Divergent series (1949; Zbl 0032.05801), Ch. 12]; the author provides a careful and full account of Ramanujan’s interesting if less than rigorous approach. Chapter 7, Sums of Powers, Bernoulli Numbers, and the Gamma Function represents a continuation of the material in Chapter 5 and 6. Ramanujan’s work on new and interesting analogs of the gamma function is given in Chapter 8. Chapter 9, Infinite Series Identities, Transformations, and Evaluations has numerous delightful results, for example, \[ if\quad f(x)=\sum^{\infty}_{k=1}h_ k x^{2k-1}/(2k-1)\quad and\quad h_ n=\sum^{n}_{k=1}1/(2k-1), \]

\[ then\quad f(1/3)=\pi^ 2/24-(1/8) Log^ 22,\quad f(1/\sqrt{5})=\pi^ 2/20,\quad and\quad f(\sqrt{5}-2)=\pi^ 2/30-(3/8) Log^ 2((\sqrt{5}-1)/2). \] The book closes with Ramanujan’s Quarterly Reports. These were prepared by Ramanujan in 1913 when he held a scholarship at the University of Madras. While there is some overlap with other chapters in this book, there is a great deal of new material in the Reports, much of it related to the theory of integral transforms.

This is a fascinating, unique book. It should greatly interest analysts, combinatorists and number theorists.

Chapter 1, Magic Squares, is probably representative of very early work and does not contain anything particularly surprising. Chapter 2, Sums related to the Harmonic Series or the Inverse Tangent Function, is also elementary; the formulas are still intriguing, e.g. \[ \sum^{\infty}_{k=0}\arctan (2/(n+2k+1)^ 2)= \arctan (1/n). \] Chapter 3, Combinatorial Analysis and Series Inversions, contains some very important results including alternatives to Lagrange inversion (see especially Entries 10 and 15); here we find that the author has gone to considerable trouble to provide rigorous corrected versions of Ramanujan’s results. In Chapter 4, Iterates of the Exponential Function and an Ingenious Formal Technique, we find the formal identity \[ \int^{\infty}_{0}x^{n-1}\quad \sum^{\infty}_{k=0}\phi (k) (- x)^ k dx=\pi \phi (-n)/\sin \pi n,\quad n>0; \] a rigorous discussion of this and related results is given. Chapter 5, Eulerian Polynomials and Numbers, Bernoulli Numbers, and the Riemann Zeta-Function, again provides a mathematically rigorous discussion of some interesting but unclear assertions by Ramanujan. The material in Chapter 6, Ramanujan’s Theory of Divergent Series, has been discussed previously by G. H. Hardy [Divergent series (1949; Zbl 0032.05801), Ch. 12]; the author provides a careful and full account of Ramanujan’s interesting if less than rigorous approach. Chapter 7, Sums of Powers, Bernoulli Numbers, and the Gamma Function represents a continuation of the material in Chapter 5 and 6. Ramanujan’s work on new and interesting analogs of the gamma function is given in Chapter 8. Chapter 9, Infinite Series Identities, Transformations, and Evaluations has numerous delightful results, for example, \[ if\quad f(x)=\sum^{\infty}_{k=1}h_ k x^{2k-1}/(2k-1)\quad and\quad h_ n=\sum^{n}_{k=1}1/(2k-1), \]

\[ then\quad f(1/3)=\pi^ 2/24-(1/8) Log^ 22,\quad f(1/\sqrt{5})=\pi^ 2/20,\quad and\quad f(\sqrt{5}-2)=\pi^ 2/30-(3/8) Log^ 2((\sqrt{5}-1)/2). \] The book closes with Ramanujan’s Quarterly Reports. These were prepared by Ramanujan in 1913 when he held a scholarship at the University of Madras. While there is some overlap with other chapters in this book, there is a great deal of new material in the Reports, much of it related to the theory of integral transforms.

This is a fascinating, unique book. It should greatly interest analysts, combinatorists and number theorists.

Reviewer: G.E.Andrews

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11-03 | History of number theory |

01A60 | History of mathematics in the 20th century |

01A75 | Collected or selected works; reprintings or translations of classics |

33-03 | History of special functions |

11Axx | Elementary number theory |

05-03 | History of combinatorics |