On the partition functions of higher spin black holes

We reconsider black hole solutions of D = 3 higher-spin gravity in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathfrak{h}\mathfrak{s}\left[ \lambda \right]\oplus \mathfrak{h}\mathfrak{s}\left[ \lambda \right] $\end{document} Chern-Simons formulation. A suitable generalisation of the BTZ black hole has a spin-3 chemical potential α, and non-zero values of all the conserved charges associated with the asymptotic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathcal{W}}_{\infty }}\left[ \lambda \right] $\end{document} symmetry. We extend the available perturbative expansion of the partition function to order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{O}\left( {{\alpha^{18 }}} \right) $\end{document} for generic values of the λ parameter. The result matches the CFT prediction at λ = 0 and at λ = 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 λ → ∞ and we derive the exact analytical expressions of the partition function and the spin-4 charge in closed form as functions of α. Also, the first subleading correction at large λ is shown to be simply related to the leading contribution.