A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $\mathcal D$ regarded as an on complex plane $\mathbb C$ and conjugate $\bar{\mathcal D}$. These operators formally commute are adjoint one to another with respect appropriate weight. find conditions when they in Nelson sense write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of Jacobi transform (synonyms: generalized Mehler–Fock transform, Olevskii transform). also show that inverse decomposition for pair commuting difference defined terms shifts imaginary direction.