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