Monotonicity properties of the zeros of Freud and sub-range Freud polynomials: Analytic and empirical results

Abstract:

Freud and sub-range Freud polynomials are orthogonal with respect to the weight function $w(t)=|t|^\mu \exp (-|t|^\nu )$, $\mu >-1$, $\nu >0$, supported on the whole real line $\mathbb {R}$, resp. on strict subintervals thereof. The zeros of these polynomials are studied here as functions of $\nu$ and shown, analytically and empirically by computation, to collectively increase or decrease on appropriate intervals of the variable $\nu$.