The Combinatorics of the leading root of Ramanujan’s function

When/Where:

March 20, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

I consider the leading root \(x_0(q)\) of Ramanujan’s function (or \(q\)-Airy 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 also obtained for the leading roots of the partial Theta function and the Painleve Airy function.