Song Heng Chan

Generalized Lambert Series Identities

We present two infinite families of generalized Lambert series identities, and deduce several known identities from them. They include an identity due to M. Jackson, a corollary of Ramanujans

Cranks and dissections in Ramanujan's lost notebook

_1\psi_1

Ramanujan and Cranks

-summation formula, and a recent identity of G. E. Andrews, R. P. Lewis and Z.-G. Liu.

The rank and crank of partitions modulo 3

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.

An Elementary Proof of Jacobi's Six Squares Theorem

The existence of the crank was first conjectured by F. J. Dyson in 1944 and was later established by G. E. Andrews and F. C. Garvan in 1987. However, much earlier, in his lost notebook, Ramanujan studied the generating function F a (q) for the crank and offered several elegant claims about it, although it seems unlikely that he was familiar with all the combinatorial implications of the crank. In particular, Ramanujan found several congruences for F a (q) in the ring of formal power series in the two variables a and q. An obscure identity found on page 59 of the lost notebook leads to uniform proofs of these congruences. He also studied divisibility properties for the coefficients of F a (q) as a power series in q. In particular, he provided ten lists of coefficients which he evidently thought exhausted these divisibility properties. None of the conjectures implied by Ramanujans tables have been proved.

Analogs of the Stern Sequence

In this paper, we prove formulas for the generating functions for the rank and crank differences for partitions modulo 3. In 2000, Andrews and Lewis made conjectures on inequalities satisfied by ranks and cranks modulo 3. These conjectures were first proved by Bringmann and Kane, respectively, using the circle method. Working directly on the generating functions, we obtain improvements to these inequalities.

Dissections of quotients of theta-functions

(2004). An Elementary Proof of Jacobis Six Squares Theorem. The American Mathematical Monthly: Vol. 111, No. 9, pp. 806-811.

Domb's numbers and Ramanujan–Sato type series for 1/π

Abstract We present two infinite families of sequences that are analogous to the Stern sequence. Sequences in the first family enumerate the set of positive rational numbers, while sequences in the second family enumerate the set of positive rational numbers with either an even numerator or an even denominator.

The odd moments of ranks and cranks

We prove a general theorem on dissections of quotients of theta-functions. As corollaries, we establish six q -series identities that were conjectured by M.D. Hirschhorn: