Michael D. Hirschhorn

Cubic analogues of the Jacobian theta function θ(z, q)

There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function 2 F 1 (1/3, 2/3 1;.) analogous to the classical θ 2 (q), θ 3 (q), θ 4 (q) and the hypergeometric function 2 F 1 (1/2 1). We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identifies are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity

Arithmetic properties of partitions with even parts distinct

AbstractIn this work, we consider the function pod(n), the number of partitions of an integer n wherein the odd parts are distinct (and the even parts are unrestricted), a function which has arisen in recent work of Alladi. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for pod(n) including the following infinite family of congruences: for all α≥0 and n≥0,

On recent congruence results of Andrews and Paule for broken k-diamonds

\mathrm{pod}\biggl(3^{2\alpha+3}n+\frac{23\times3^{2\alpha+2}+1}{8}\biggr)\equiv 0\ (\mathrm{mod}\ 3).

On representations of a number as a sum of three squares

One of Ramanujans unpublished, unproven identities has excited considerable interest over the years. Indeed, no fewer than four proofs have appeared in the literature. The object of this note is to present yet another proof, simpler than the others, relying only on Jacobis triple product identity.

On the Expansion of Ramanujan's Continued Fraction

In one of their most recent works, George Andrews and Peter Paule continue their study of partition functions via MacMahon’s Partition Analysis by considering partition functions associated with directed graphs which consist of chains of hexagons. In the process, they prove a congruence related to one of these partition functions and conjecture a number of similar congruence results. Our first goal in this note is to reprove this congruence by explicitly finding the generating function in question. We then prove one of the conjectures posed by Andrews and Paule as well as a number of congruences not mentioned by them. All of our results follow from straightforward generating function manipulations.

Equilateral convex pentagons which tile the plane

Using elementary means, we derive an explicit formula for a3.n/, the number of 3-core partitions of n, in terms of the prime factorization of 3nC 1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3.n/, one of which specializes to the recent result of Baruah and Berndt which states that, for all n 0, a3.4nC 1/D a3.n/. 2000 Mathematics subject classification: primary 05A17, 11P81, 11P83.

Ramanujan's "Most Beautiful Identity"

In a manuscript discovered in 1976 by George E. Andrews, Ramanujan states a formula for a certain continued fraction, without proof. In this paper we establish formulae for the convergents to the continued fraction, from which Ramanujans result is easily deduced.