Zero distribution of orthogonal polynomials

Abstract

In [5], Karlin and Szego conjectured : If ${P_n(x)}^∞_{n=0}$ is an orthogonal polynomial system and ${P``_{n}(x)}^∞_{n=1}$ is a Sturm sequence, then ${P_n(x)}^∞_{n=0}$ is essentially (that is, after a linear change of variable) a classical orthogonal polynomial system of Jacobi, Laguerre, or Hermite. Here, we prove that for any orthogonal polynomial system ${P_n(x)}^∞_{n=0}$, ${P``_{n}(x)}^∞_{n=1}$ is always a Sturm sequence. Thus, in particular, the above conjecture by Karlin and Szeg$ö$ is false. And zero distribution of the polynomials ${S_{n}(x)}^∞_{n=0}$ which are orthogonal with respect to the discrete Sobolev inner product type. ＜f,g> = ∫_If(x)g(x)dμ(x)+$Nf^{(r)}(c)g^{(r)}(c)$ where I is a real interval, with I = [a,b], a,b ∈ R, N ≥ 0, r ≥ 1, c ≥ b and let μ(x) be an arbitrary distribution function on I. $S_{n}(x)$ has n real, simple zeros. The location of these zeros is given in relation to the position of the zeros of some classical polynomials (i.e., polynomials with respect to an inner product with N=0).