Heng Huat Chan

Ramanujan's Series for 1/π: A Survey ∗

When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. 1. THE BEGINNING. Toward the end of the first paper [57], [58 ,p . 36] that Ramanujan published in England, at the beginning of Section 13, he writes, “I shall conclude this paper by giving a few series for 1/π.” (In fact, Ramanujan concluded his paper a couple of pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanujan records three series representations for 1/π .A s is customary, set

The Rogers-Ramanujan continual fraction

Abstract A survey of many theorems on the Rogers–Ramanujan continued fraction is provided. Emphasis is given to results from Ramanujans lost notebook that have only recently been proved.

Ramanujan and the modular j-invariant

A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujans assertions about tn by establishing new connections between the modular j- invariant and Ramanujans cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of Q( p n). This shows that tn is a new class invariant according to H. Webers definition of class invariants.

Some values for the Rogers-Ramanujan continued fraction

In his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujans values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujans beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions. 0. Introduction. Let, for \q\ < 1, n, x a / q q q and S(q) = -R(-q) denote the famous Rogers-Ramanujan continued fractions. In both his first and second letters to Hardy [11, pp. xxvii, xxviii], Ramanujan communicated theorems about R(q) and S(q). In particular, in his first letter, he asserted that

Rational analogues of Ramanujan's series for 1/π

A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

CONGRUENCES SATISFIED BY APÉRY-LIKE NUMBERS

HENG HUAT CHAN∗, SHAUN COOPER† and FRANCESCO SICA‡ ∗Department of Mathematics, National University of Singapore Block S17, 10, Lower Kent Ridge Road, 119076 Singapore matchh@nus.edu.sg †Institute of Information and Mathematical Sciences Massey University, Private Bag 102904 North Shore Mail Centre, Auckland, New Zealand s.cooper@massey.ac.nz ‡Mathematics and Computer Science, Mount Allison University 67 York Street, Sackville, NB, E4L 1E6, Canada fsica@mta.ca

NEW REPRESENTATIONS FOR APÉRY-LIKE SEQUENCES

We prove algebraic transformations for the generating series of three Apery-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method that derives three new series for 1= from a classical Ramanujans series.

Cranks and dissections in Ramanujan's lost notebook

In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F.J. Dyson and later established by G.E. Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating function F(q) of cranks. All are found, sometimes in abbreviated form, in the lost notebook, and imply dissections of F(q). Consequences of our work are interesting new q-series identities and congruences in the spirit of Atkin and Swinnerton-Dyer.

Ramanujan's Singular Moduli

In his first notebook, in scattered places, Ramanujan recorded without proofs the values of over 100 class invariants and over 30 singular moduli. This paper is devoted to establishing all of Ramanujans representations for singular moduli. For those of odd index, new algorithms needed to be developed.