Factorizations of Theta Function Identities

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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.

mailto:s.cooper@massey.ac.nz

Heung Yeung Lam

Institute of Natural and Mathematical Sciences

Massey University-Albany

Private Bag 102904, North Shore Mail Centre

Auckland, New Zealand.

mailto:H.Y.Lam@massey.ac.nz




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

\phi(q) + \phi(q^5),\quad \phi(q)-\phi(q^5),\quad\phi(q)+\sqrt{5}\phi(q^5)\quad\mbox{and}\quad \phi(q)-\sqrt{5}\phi(q^5)

where \phi(q) = \sum_{n=-\infty}^\infty q^{n^2} 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

\phi(q^3) \pm \phi(q),\quad \phi(q^3)\pm i\,\phi(q),\quad \sqrt{3}\,\phi(q^3) \pm i\,\phi(q)\quad\mbox{and}\quad \sqrt{3}\phi(q^3)\pm \phi(q)

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

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