Department of Mathematics
Pennsylvania State University
The Scariest Formulas in Ramanujan's Lost Notebook
I have spent the last
twenty eight years on and off studying Ramanujan's Lost Notebook.
For most of this time I worked hard at ignoring two formulas neatly
buried in amongst other results that were much easier for me to
handle. One of these formulas provides an infinite product
representation for the two variable Rogers-Ramanujan series and the
other provides a comparable product for the partial theta series.
In this talk I shall attempt to provide a comprehensible account of how
the proofs of these results were obtained. On the surface each
relies on the theory of entire functions and Hadamard's famous product
representation of entire functions. However, much more detailed
information is required for a complete understanding of Ramanujan's
discoveries. The road to enlightenment starts with the
Stieltjes-Wigert polynomials and an old paper of Szeg\"o.