Asymptotic structure of eigenvalues and eigenvectors of certain triangular Hankel matrices

The Hankel matrices considered in the article arose at one reformulation of the Riemann hypothesis proposed earlier by the author.
Computer calculations showed that in the case of the Riemann zeta function the eigenvalues and the eigenvectors of such matrices have an interesting structure.
The article studies a model situation when instead of the zeta function function one takes a function having a single zero. For this case we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors.