The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

Abstract

dedicated to jaap korevaar on his 75th birthdayWe study the zeros of orthogonal polynomials p n, N , n=0, 1, ..., that are generated by recurrence coefficients a n, N and b n, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio nÂN converges, we show that the polynomials p n, N have an asymptotic zero distribution as nÂN tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distributions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes Wigert polynomials, and Lommel polynomials.

Integrable open spin chains related to infinite matrix product states

“…To begin with, it is shown in Ref. 27 that when N → ∞ the density of the zeros u j of the Jacobi polynomial P…”

Density of zeros of some orthogonal polynomials

“…This derivation again is based on the thermodynamic approach of [4]. The above argument does not replace the original proof of the theorem [10]. Regarding the accuracy of the Coulomb fluid approach, we have the following quote:…”

“…We note here that the zero density plays an important role in determining the strong asymptotics of orthogonal polynomials and these in turn are essential ingredients in determining fundamental physical quantities such as the gap formation probability [4,6]. Recently, Kuijlaars and Van Assche [10] proved that under some mild conditions imposed on the (varying) recurrence coefficients, the zero density can be computed by quadrature. In this situation, the input consisting of the recurrence coefficients therefore is independent of the weight.…”

“…This proves to be a complementary tool for obtaining the density and provides an independent check on the Coulomb fluid technique. We consider various examples of orthogonal polynomials with varying recurrence coefficients, and then, utilizing a theorem given in [10], stated below, the density is derived. In the case of Stieltjes-Wigert polynomials, we show that the density calculated by this method is identical to that found by the Coulomb fluid approach.…”

“…Note that Pn,N(x) defined above is monic and (1.3) was derived as thermodynamic relations of Hermitian random matrix models [4]. Although (1.4) is proved by starting from the recurrence relations [10], we give below a heuristic justification of how it can be obtained using the Coulomb fluid approach. In the potential theoretic/Coulomb fluid approach, with varying weights, that is v(x) -► Nv(x) where 0 < N < oo, the zero density, denoted by cr(x,t), satisfies the following integral equation…”