On a ${\mathbb{C}}^2$-valued integral index transform and bilateral hypergeometric series

We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{R}\mathrm{e}z=1\mathrm{/}2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group $SL(2\mathbb{R}$. Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula.