Asymptotics of recurrence coefficients for orthonormal polynomials on the line - Magnus's method revisited

We use Freud equations to obtain the main term in the asymptotic expansion of the recurrence coefficients associated with orthonormal polynomials pn(w2) for weights w = W exp(-Q) on the real line where Q is an even polynomial of fixed degree with nonnegative coefficients or where Q(x) -- exp(x2m), m ≥ 1. Here W(x) = |x|ρ for some real ρ > -1.

[1]  Pavel Bleher,et al.  Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model , 1999, math-ph/9907025.

[2]  P. Nevai,et al.  Asymptotics for solutions of systems of smooth recurrence equations. , 1988 .

[3]  E. A. Rakhmanov.  Strong asymptotics for orthogonal polynomials , 1993 .

[4]  Weak convergence of orthogonal polynomials , 1994, math/9401209.

[5]  W. Fleming.  Functions of Several Variables , 1965 .

[6]  Hrushikesh Narhar Mhaskar,et al.  Introduction to the theory of weighted polynomial approximation , 1997, Series in approximations and decompositions.

[7]  Vilmos Totik,et al.  Weighted Approximation with Varying Weight , 1994 .

[8]  Max L. Warshauer,et al.  Lecture Notes in Mathematics , 2001 .

[9]  Géza Freud,et al.  On the greatest zero of an orthogonal polynomial , 1986 .

[10]  Stephanos Venakides,et al.  Strong asymptotics of orthogonal polynomials with respect to exponential weights , 1999 .

[11]  A. Magnus.  On Freud's equations for exponential weights , 1986 .

[12]  Alphonse P. Magnus,et al.  A Proof of Freud Conjecture About the Orthogonal Polynomials Related To [x]rho-exp[-x2m], for Integer-m , 1985 .

[13]  A. Magnus.  Symmetries and Integrability of Difference Equations: Freud's equations for orthogonal polynomials as discrete Painlevé equations , 1996, math/9611218.

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  Doron S. Lubinsky,et al.  Orthogonal Polynomials for Exponential Weights , 2001 .

[16]  H. Mhaskar,et al.  A proof of Freud's conjecture for exponential weights , 1988 .