Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces

We discuss properties of the index hypergeometric transform (it is named also the Jacobi transform or the Olevsky transform) interpolating analysis of Berezin kernels on rank 1 symmetric spaces. We discuss a unitary intertwining operator from $L^2$ on symmetric space to Berezin deformation of $L^2$. We also find images of some differential operators under the index transform.