Refined Verlinde and Segre formula for Hilbert schemes

Let $\mathrm{Hilb}_nS$ be the Hilbert scheme of $n$ points on a smooth projective surface $S$. To a class $\alpha\in K^0(S)$ correspond a tautological vector bundle $\alpha^{[n]}$ on $\mathrm{Hilb}_nS$ and line bundle $L_{(n)}\otimes E^{\otimes r}$ with $L=\det(\alpha)$, $r=\mathrm{rk}(\alpha)$. In this paper we give closed formulas for the generating functions for the Segre classes $\int_{\mathrm{Hilb}_nS} s(\alpha^{[n]})$, and the Verlinde numbers $\chi(\mathrm{Hilb}_nS,L_{(n)}\otimes E^{\otimes r})$, for any surface $S$ and any class $\alpha\in K^0(S)$. In fact we determine a more general generating function for $K$-theoretic invariants of Hilbert schemes of points, which contains the formulas for Segre and Verlinde numbers as specializations. We prove these formulas in case $K_S^2=0$. Without assuming the condition $K_S^2=0$, we show the Segre-Verlinde conjecture of Johnson and Marian-Oprea-Pandharipande, which relates the Segre and Verlinde generating series by an explicit change of variables.