On orthogonal polynomials with positive zeros and the associated kernel polynomials (Q1076270)

Let \(\{P_ n(x)\}_ n\) be a sequence of monic polynomials orthogonal with respect to a distribution on the nonnegative real axis. Three aspects of such sequences are discussed in this paper, viz., (i) the coefficients in the three term recurrence relation, (ii) the behaviour of \(P_ n(x)/P_ n(0)\) as n goes to infinity, and (iii) the spectrum of the distribution. A number of theorems connecting these aspects are given. The kernel polynomials associated with a sequence \(\{P_ n(x)\}\) play an important role in proving these theorems.