  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}_{+}) $.
