On the partition functions of higher spin black holes

We reconsider black hole solutions of D = 3 higher-spin gravity in the $hs[\lambda]\oplus hs[\lambda]$ Chern-Simons formulation. A suitable generalisation of the BTZ black hole has a spin-3 chemical potential $\alpha$, and non-zero values of all the conserved charges associated with the asymptotic $\mathcal W_\infty[\lambda]$ symmetry. We extend the available perturbative expansion of the partition function to order $\mathcal O(\alpha^{18})$ for generic values of the $\lambda$ parameter. The result matches the CFT prediction at $\lambda=0$ and at $\lambda=1$ where we provide the exact all-order expansion of the partition function. The perturbative series is then analysed in the interesting non-trivial limit $\lambda\to\infty$ and we derive the exact analytical expressions of the partition function and the spin-4 charge in closed form as functions of $\alpha$. Also, the first subleading correction at large $\lambda$ is shown to be simply related to the leading contribution.