Associated Polynomials and Birth-Death Processes.

摘要

We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials, with a view to applications in the setting of birth-death processes. In particular, we relate the supports of the two measures, and their moments of positive and negative orders. Our results indicate how the prevalence of recurrence or a-recurrence in a birth-death process can be recognized from certain properties of an associated measure.