Discrete Inequalities, Orthogonal Polynomials and the Spectral Theory of Difference Operators

In a recent paper the first two authors studied a class of series inequalities associated with a three-term recurrence relation which includes a well-known inequality of Copson's. It was shown that the validity of the inequality and the value of the best constant are determined in terms of the so-called Hellinger-Nevanlinna m-function. The theory is the discrete analogue of that established by Everitt for a class of integro-differential inequalities. In this paper the properties of the m-function are investigated and connections with the theory of orthogonal polynomials and the Hamburger moment problem are explored. The results are applied to give examples of the series inequalities associated with the classical orthogonal polynomials.