The Eigenfunctions of the Hilbert Matrix

For each noninteger complex number ??, the Hilbert matrix $$H_lambda= biggl( frac{1}{n+m+lambda} biggr)_{n,mgeq0}$$ defines a bounded linear operator on the Hardy spaces $mathcal{H}^{p}$ , 1<p $mathcal{A}^{-tau}$ , ??>0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex ?? results by Hill and Rosenblum for real ??. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.