Verlinde series for Hirzebruch surfaces

We give an explicit formula for Euler characteristics of line bundles on the Hilbert scheme of points on $\mathbb{P}^1\times\mathbb{P}^1$. Combined with structural results of Ellingsrud, G\"ottsche, and Lehn, this determines the Euler characteristic of any line bundle on the Hilbert scheme of points on any smooth, projective surface. We also give an enumerative description of the dimensions of spaces of global sections of ample line bundles on Hilbert schemes of points on Hirzebruch surfaces, extending the polytope-line bundle correspondence on the underlying toric surface.