  A complete characterization is provided of Hankel matrices commuting with
Jacobi matrices which correspond to hypergeometric orthogonal polynomials from
the Askey scheme. It follows, as the main result of the paper, that the
generalized Hilbert matrix is the only prominent infinite-rank Hankel matrix
which, if regarded as an operator on $\ell^{2}(\mathbb{N}_{0})$, is
diagonalizable by application of the commutator method with Jacobi matrices
from the mentioned families.
