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We introduce and study invariant differential operators acting on the space$\mathcal{H}(\Omega)$ of holomorphic functions on the complement${\Omega=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the"complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w=1\}$. We obtain recursion identities, describe the behaviour under change ofcoordinates and find the generators of the corresponding operator algebra. Weillustrate how this provides a unified framework for investigating conformallyinvariant differential operators on the unit disk $\mathbb{D}$ and the Riemannsphere $\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Mindaand many others, within their conjecturally natural habitat. We apply themachinery to a problem in deformation quantization by deriving explicitformulas for the canonical Wick-type star products on $\Omega$, the unit disk$\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of suchinvariant differential operators. These formulas are given in form of factorialseries which depend holomorphically on a complex deformation parameter $\hbar$and lead to asymptotic expansions of the star products in powers of $\hbar$.

Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond