Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Abstract

We consider orthogonal polynomials {P n,N , n = 0, 1, 2, . . .}, where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit W = lim N→∞ W 1/N N and link this limit to an external field for an equilibrium problem in logarithmic potential theory.

Measures for orthogonal polynomials with unbounded recurrence coefficients

“…, An extension of this formula was used in [1] to show the connection between the limit of varying recurrence coefficients and their corresponding orthogonality measure and an analog of this formula for the unbounded case is necessary in order to extend the above results to the unbounded case. This is done in the next section.…”

Asymptotic behavior of the spectrum of combination scattering at Stokes phonons

“…Такие системы рассматривались в работах (см. работы [13]- [16], рекуррентные соотношения более высокого порядка рассматривались в [17]- [19]).…”

Asymptotics for multiple Meixner polynomials

“…The proof of this Lemma follows from Poincare's theorem on ratio asymptotics of solutions of the recurrence relations (see [20]) and from the theorem of Kuijlaars and Van Assche on the nth root asymptotics of the solutions of the recurrence relations with varying coefficients (see [13] as well as [5] and [6]).…”