# Generating functions, Tutte polynomials, and the bivariate series $\sum_n x^n y^{n^2} / n!$.

A few years ago I computed the Tutte polynomials of the matroids given by the classical Coxeter groups, and found that their generating functions are all simple variations of the series $\sum_n \frac{x^n y^{n^2}}{n!}$. I've wondered if there is a more geometric/algebraic explanation of this. Is this series known? Are there other natural occurrences of it that might be relevant?

-
Don't you mean $\sum_n x^n y^{n^2}/n!$? –  Ira Gessel Jul 3 '12 at 13:39
Is it known whether $\sum x^n y^{n^2} / {n!}$ satisfies an ADE? –  Martin Rubey Jul 3 '12 at 17:03
The generating function $f(x) = \sum_n x^n y^\binom{n}{2}/n!$ satisfies $f'(x) = f(xy)$. –  Ira Gessel Jul 3 '12 at 17:17
So writing $g(u)=f(\exp(u))$, Ira's equation becomes a "delay differential equation" where the derivative $g'(u)$ is written in terms of $g(u-\tau)$, for some $\tau$. –  Gerald Edgar Jul 3 '12 at 18:49