Hankel and Toeplitz operators: continuous and discrete representations

We find a relation guaranteeing that Hankel operators realized in the space of sequences $\ell^2 ({\Bbb Z}_{+})$ and in the space of functions $L^2 ({\Bbb R}_{+})$ are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space $\ell^2 ({\Bbb Z}_{+})$ generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces $\ell^2 ({\Bbb Z}_{+})$ and $L^2 ({\Bbb R}_{+})$.