The variation of zeros of certain orthogonal polynomials
The author considers certain orthogonal polynomial systems defined by a three-term recurrence relation in which the coefficients are functions of a parameter \(\nu\), and seeks conditions for the largest and smallest zeros of these polynomials to be monotone functions of \(\nu\). A typical result is that the largest zero of a birth-death polynomial increases (decreases) with \(\nu\) if the birth and death rates are increasing (decreasing) functions of \(\nu\). Related results are obtained for random walk polynomials and a class of polynomials which includes the Lommel polynomials. The proofs use a finite-dimensional version of the Hellman- Feynman theorem.
scientific article; zbMATH DE number 4021605
scientific article