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

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