The problem of moments on a finite set of points

The influence of the last diagonal entry $b_n$ of the Jacobi matrix on its eigenvalues, which at the same time are the nodes of orthogonality of respective polynomials as well as on the squares of the first components of the normalized eigenvectors – the weights of the orthogonality, is considered. The weights of orthogonality are the distribution masses whose moments are known and given by the positive definite Hankel matrix independent of $b_n$. Using the solutions to the equations with special matrices the first derivatives of $b_n$ of the nodes and the weights of orthogonality of the polynomials are calculated. Their asymptotic behavior with $b_n\to\pm\infty$ is discussed.