The Hilbert matrix done right

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$H={\left(\frac{1}{i+j+1}\right)}_{i,j\ge 0}$$ which defines a bounded linear operator on the sequence space ${\mathrm{\ell }}^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,\pi ]$ with no eigenvalues (Magnus’ result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill’s result).