# Factorizations of Theta Function Identities

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 47, (2017), **Pages** 9-21

**Shaun Cooper**

Institute of Natural and Mathematical Sciences

Massey University-Albany

Private Bag 102904, North Shore Mail Centre

Auckland, New Zealand.

**Heung Yeung Lam **

Institute of Natural and Mathematical Sciences

Massey University-Albany

Private Bag 102904, North Shore Mail Centre

Auckland, New Zealand.

**Abstract** In Ramanujan's lost notebook, infinite product formulas are recorded for
each of the functions

where is the generating function for squares. Ramanujan also gave similar results that involve the Rogers--Ramanujan continued fraction. We provide a survey of these and other identities. We state and prove cubic analogues of Ramanujan's results, many of which are new. That is, we provide factorizations for the eight functions

as well as corresponding results for the generating function of the triangular numbers.

**Keywords** Dedekind eta function, infinite product, Jacobi triple product identity, Rogers--Ramanujan continued fraction, theta function.

**Classification** (MSC2000) Primary 11F11. Secondary 05A19, 11P83