The Combinatorics of the leading root of Ramanujan’s function

When/Where:

August 21, 2015, 1:55 — 2:45pm at LIT 368.

Abstract:

I consider the leading root \(x_0(q)\) of Ramanujan’s function
\(\sum\limits_{n=0}^\infty\frac{(-x)^nq^{n^2}}{(1-q)(1-q^2)\ldots(1-q^n)}\).
I prove that its formal power series expansion
\(qx_0(-q)=1+q+q^2+2q^3+4q^4+8q^5+\ldots\)

has positive integer-valued coefficients, by giving an explicit combinatorial interpretation
of these numbers in terms of trees whose vertices are decorated with polyominos.

Similar results are obtained for the leading roots of the partial Theta function and the
Painleve Airy function.