##
**Ramanujan’s notebooks. Part II.**
*(English)*
Zbl 0716.11001

New York etc.: Springer-Verlag. xi, 359 p. DM 168.00 (1989).

Let me begin with an extended excerpt from George Andrews’ review of Part I of this series of books (1985) (there will be four in all when Berndt finishes) in Zbl 0555.10001. “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 with rigorous proofs (seldom even hinted at in the original Notebooks) and relevent references. This is a beautiful job, and every mathematician owes Bruce Berndt warm thanks.”

Berndt has been at work on this project for more than a decade. His patience, precision, thoroughness and dedication are evident on each page. A typical entry has the following form: Ramanujan’s words are given in italics. Proofs and discussion follow in roman font. The “relevent references” alluded to above range from the mid-18th Century to the day before yesterday. The discussions are lively and offer glimpses into the frustrations - and rewards - of this is endeavor. More on this below. As one would expect, obtaining proofs of Ramanujan’s results not already in the literature required collaborators. There is a table in the Introduction listing these previously published works; R. Evans and R. Lamphere were the main contributors. As Berndt remarks, these proofs, which rely on function theory, must be quite different from those of Ramanujan.

As to the contents, the Introduction contains a fine and careful summary. Then, each chapter has a head note giving a more precise summary and pointing to highlights contained therein. The second Notebook is an enlarged edition of the first. Part I of the Berndt’s series covered Chapters 1-9 of this second Notebook. In Part II we have Chapters 10-15. Here are the chapter titles: Chapter 10 is Hypergeometric Series I. Chapter 11 is Hypergeometric Series II. Chapter 12 is Continued Fractions (the analytic theory - reviewer). Chapter 13 is Integrals and Asymptotic Expansions. Chapter 14 is Infinite Series and Chapter 15 is Asymptotic Expansions and Modular Forms (warning: Ramanujan’s definition of modular equation is not the standard one). Ramanujan gave no Chapter titles, so Berndt has supplied them. Main Entries are very often followed by Corollaries and Examples. As Berndt states, “Ramanujan possessed the uncanny ability for finding the most important examples of theorems,...”. Here Berndt shows his flair for understatement! These examples are exquisite, gorgeous. If you love Euler, you’ll love these jewels.

As Berndt and others have noted, Ramanujan probably borrowed his expository style - notably the nearly complete suppression of proof - from the only high level text he studied before leaving for England, Carr’s ‘A Synopsis of Elementary Results in Pure Mathematics’. This has left Berndt with a job with aspects sometimes akin to that of a mathematical archeologist: One turns an object over and over and tries to deduce what was in the mind of the maker. The fact that the maker in this case was an Olympian only amplifies the situation. Here two examples of this process that intrigued the reviewer. Berndt’s Introduction tells us that Entry 24 of Hypergeometric Series II (Chapter 11) is “By far, the most interesting... (in the Chapter)”. The paragraph following the statement of Corollary 2 of this entry begins thus: “We cannot see how Corollary 2 would follow from Entry 24.” How many man/weeks of frustration are implied by this sentence?

This volume marks the half-way point of Berndt’s edition of the Ramanujan Notebooks. Volume Three is in its final stages at Springer-Verlag. One cannot but marvel at the consistant quality here; and all wish him - and his co-workers - Godspeed as they move nearer completion of this noble effort.

Berndt has been at work on this project for more than a decade. His patience, precision, thoroughness and dedication are evident on each page. A typical entry has the following form: Ramanujan’s words are given in italics. Proofs and discussion follow in roman font. The “relevent references” alluded to above range from the mid-18th Century to the day before yesterday. The discussions are lively and offer glimpses into the frustrations - and rewards - of this is endeavor. More on this below. As one would expect, obtaining proofs of Ramanujan’s results not already in the literature required collaborators. There is a table in the Introduction listing these previously published works; R. Evans and R. Lamphere were the main contributors. As Berndt remarks, these proofs, which rely on function theory, must be quite different from those of Ramanujan.

As to the contents, the Introduction contains a fine and careful summary. Then, each chapter has a head note giving a more precise summary and pointing to highlights contained therein. The second Notebook is an enlarged edition of the first. Part I of the Berndt’s series covered Chapters 1-9 of this second Notebook. In Part II we have Chapters 10-15. Here are the chapter titles: Chapter 10 is Hypergeometric Series I. Chapter 11 is Hypergeometric Series II. Chapter 12 is Continued Fractions (the analytic theory - reviewer). Chapter 13 is Integrals and Asymptotic Expansions. Chapter 14 is Infinite Series and Chapter 15 is Asymptotic Expansions and Modular Forms (warning: Ramanujan’s definition of modular equation is not the standard one). Ramanujan gave no Chapter titles, so Berndt has supplied them. Main Entries are very often followed by Corollaries and Examples. As Berndt states, “Ramanujan possessed the uncanny ability for finding the most important examples of theorems,...”. Here Berndt shows his flair for understatement! These examples are exquisite, gorgeous. If you love Euler, you’ll love these jewels.

As Berndt and others have noted, Ramanujan probably borrowed his expository style - notably the nearly complete suppression of proof - from the only high level text he studied before leaving for England, Carr’s ‘A Synopsis of Elementary Results in Pure Mathematics’. This has left Berndt with a job with aspects sometimes akin to that of a mathematical archeologist: One turns an object over and over and tries to deduce what was in the mind of the maker. The fact that the maker in this case was an Olympian only amplifies the situation. Here two examples of this process that intrigued the reviewer. Berndt’s Introduction tells us that Entry 24 of Hypergeometric Series II (Chapter 11) is “By far, the most interesting... (in the Chapter)”. The paragraph following the statement of Corollary 2 of this entry begins thus: “We cannot see how Corollary 2 would follow from Entry 24.” How many man/weeks of frustration are implied by this sentence?

This volume marks the half-way point of Berndt’s edition of the Ramanujan Notebooks. Volume Three is in its final stages at Springer-Verlag. One cannot but marvel at the consistant quality here; and all wish him - and his co-workers - Godspeed as they move nearer completion of this noble effort.

Reviewer: M.Sheingorn

### 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 |

33-03 | History of special functions |

41-03 | History of approximations and expansions |

40-03 | History of sequences, series, summability |

### Keywords:

Ramanujan’s Notebooks; Hypergeometric Series; Continued Fractions; Integrals; Asymptotic Expansions; Infinite Series; Modular Forms### Biographic References:

Ramanujan, S.### Citations:

Zbl 0555.10001
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XMLCite

\textit{B. C. Berndt}, Ramanujan's notebooks. Part II. New York etc.: Springer-Verlag (1989; Zbl 0716.11001)

### Digital Library of Mathematical Functions:

§15.7 Continued Fractions ‣ Properties ‣ Chapter 15 Hypergeometric Function### Online Encyclopedia of Integer Sequences:

Decimal expansion of -exp(1)*Ei(-1), also called Gompertz’s constant, or the Euler-Gompertz constant.Decimal expansion of Gamma”(1).

a(n) = Sum_{k=0..n-1} sigma(2k+1)*sigma_3(n-k).

a(1) = 1, a(2) = 3, a(n+2) = 3*a(n+1) + (n+1)^2*a(n).

a(1) = 1, a(2) = 7, a(n+2) = 7*a(n+1) + (n+1)^2*a(n).

a(1) = 1, a(2) = 9, a(n+2) = 9*a(n+1) + (n+1)^2*a(n).

a(1) = 1, a(2) = 2, a(n+2) = 2*a(n+1) + (n+1)*(n+2)*a(n).

a(1) = 1, a(2) = 4, a(n+2) = 4*a(n+1) + (n+1)*(n+2)*a(n).

a(1) = 1, a(2) = 6, a(n+2) = 6*a(n+1) + (n+1)*(n+2)*a(n).

a(1) = 1, a(2) = 8, a(n+2) = 8*a(n+1) + (n+1)*(n+2)*a(n).

a(1) = 1, a(2) = 10, a(n+2) = 10*a(n+1) + (n+1)*(n+2)*a(n).

a(1) = 1, a(2) = 3, a(n+2) = 3*a(n+1)+(n+1)*(n+3)*a(n).

a(0) = 0, a(1) = 1, a(n+1) = (2*n^2 + 2*n + 3)*a(n) - n^4*a(n-1), n >= 1.

a(0) = 0, a(1) = 1, a(n+1) = (2*n^2+2*n+7)*a(n) - n^4*a(n-1), n >= 1.

a(0) = 0, a(1) = 1, a(n+1) = (2*n^2+2*n+13)*a(n) - n^4*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = (2*n^2+2*n+21)*a(n) - n^4*a(n-1).

a(0) = 0, a(1) = 1; for n>1, a(n+1) = (2*n+1)*a(n) + n^4*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = 3*(2*n+1)*a(n) + n^4*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = 5*(2*n+1)*a(n) + n^4*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = 7*(2*n+1)*a(n) + n^4*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+5)*a(n) - n^6*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+13)*a(n) - n^6*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+25)*a(n) - n^6*a(n-1).

a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*(n^2+n+41)*a(n) - n^6*a(n-1).