F-signature functions of diagonal hypersurfaces

Let f be a diagonal hypersurface in Aₚ=Fₚ[[x₁,,xₙ]]. We study the behavior of the function φf,p(a/pᵉ)=p⁻ⁿᵉFₚ(Aₚ/(x₁pᵉ,,xₙpᵉ,fᵃ)) which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when p goes to infinity φf,p converges to a piecewise polynomial function φf and the left and right derivatives of φf,p converge to φ'f. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When f is a Fermat hypersurface, we investigate the shape of the F-signature function of f and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed p. This allows us to answer negatively to a question of Watanabe and Yoshida.