Assume the semi-infinite Hankel matrix $H$ with entries $$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$ where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a semi-infinite Jacobi matrix $T$ commuting (formally) with $H$ can be determined explicitly. For example, if $H$ is the Hilbert matrix, i.e. $\alpha_{n}=1/n$, then one can set $$T_{n,n}=2n(n-1), \quad T_{n,n+1}=T_{n+1,n}=-n^{2} \quad (T_{m,n}=0 \mbox{ otherwise})$$ and $HT=TH$, indeed.
My question is twofold. Is there any systematic way how to determine the diagonal and off-diagonal sequence of the Jacobi matrix $T$ which commutes with a given Hankel matrix $H$ (perhaps a computer based method determining several first entries of the sequences)?
More precisely, the question is: how to find sequences $b_{n}=T_{n,n}$ and $a_{n}=T_{n,n+1}$, such that $$(a_{j-1}-a_{i-1})\alpha_{i+j-1}+(b_{j}-b_{i})\alpha_{i+j}+(a_{j}-a_{i})\alpha_{i+j+1}=0, \quad \forall i,j\geq1,$$ where one sets $a_{0}=0$ and $\alpha_{n}\in\mathbb{R}$ is given. Of course, only non-trivial solution is of interest.
Second, is any other example of commuting Hankel and Jacobi matrix known? For instance, interesting Hankel matrices correspond to the choice $\alpha_{n}=1/n^{2}$ (or more general powers of $n$) or $\alpha_{n}=1/n!$.
Any information related to the post would be useful. Thanks.
