A Note on the Derivatives of Isotropic Positive Definite Functions on the Hilbert Sphere

In this note we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Tr"ubner and Ziegel, which says that for a positive definite function on the Hilbert sphere to be in $C^{2\mathrm{\ell }}([0,\pi ])$, it is necessary and sufficient for its $\mathrm{\infty }$-Schoenberg sequence to satisfy ${\sum }_{m=0}^{\mathrm{\infty }}{a}_m{m}^{\mathrm{\ell }}<\mathrm{\infty }$.