Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights

Abstract

Abstract.Letp"(x) = y"x" + •■ ■ denote the nth polynomial orthonormal with respect to the weight exp (-x^/ß) where ß > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing a" = t"-\/in, the expression a"n~1/P has a limit as n -» oo. It is shown that this expression has an asymptotic expansion in terms of negative even powers oí n. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.Consider the polynomials pn that are orthonormal with respect to the weight function exp(-\x\ß/ß) on the real line, where ß is a positive real number. Denoting by y" the leading coefficient of pn (n > 0) and writing an = y" i/y" for n > 1 and an = 0 for n < 0, G. Freud conjectured thatholds for every positive even ß (see [3, Conjecture, p. 5]; his conjecture has a slightly different form, as he considered the weight function |jc|pexp(-|jc|^) rather than the one above). He also entertained the possibility that this conjecture is valid for all positive real ß. In case ß > 0 is even, he proved that if the limit on the left exists then it must have the value on the right-hand side (see [3, Theorem 1 on p. 4]), and he established the conjecture for ß = 2, 4, and 6 (see [3, pp. 5-6]). He accomplished these by extracting information from the formula (2) -=\ Pn(x)p"-X(x)x-l\x\ exp -|x| /ß) dx,

Untitled

“…Nevai, and Zaslavsky [20] for positive even integers β, yielding that r (rec) β = O(n −2 ) can be expanded in an asymptotic series in terms of powers of n −2 . Motivated by applications in approximation theory, Lubinsky raised in [15] the question for which classes of weight one can prove that r (rec) β is of order O(n −1 ).…”

A Generalized Freud Weight

“…Freud explored other properties, such as the asymptotic behavior of the polynomials using the recurrence coefficients and the asymptotic behavior of the greatest zero [24]. Recent contributions on the asymptotic behavior of the recurrence coefficients associated with Freud-type exponential weights and zeros of the associated polynomials can be found in [2,35,37,41,43,44,45,50,54] Magnus [40] showed that the coefficients in the three-term recurrence relation for the Freud weight ω(x; t) = exp −x 4 + tx 2 , x, t ∈ R, with t ∈ R a parameter, can be expressed in terms of simultaneous solutions, q n , of the discrete equation q n (q n−1 + q n + q n+1 ) + 2tq n = n, (…”

Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations

“…The special positive solution needed to get the recurrence coefficients was analyzed by Nevai [32] and Lew and Quarles [26]. An asymptotic expansion was found by Máté-Nevai-Zaslavsky [30]. Only later (in 1991) it was recognized as a discrete Painlevé equation by Fokas, Its and Kitaev [17] who coined the name d-P I .…”