(C, 1) Means of Orthonormal Expansions for Exponential Weights

Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights

Let wρ x : |x|exp −Q x , ρ > −1/2, where Q ∈ C2 : −∞,∞ → 0,∞ is an even function. In 2008 we have a relation of the orthonormal polynomial pn w2 ρ;x with respect to the weightw 2 ρ x ; p′ n x An x pn−1 x − Bn x pn x − 2ρnpn x /x, where An x and Bn x are some integrating functions for orthonormal polynomials pn w2 ρ;x . In this paper, we get estimates of the higher derivatives of An x and Bn x ,...

Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Let R −∞,∞ , and let Q ∈ C2 : R → 0,∞ be an even function. In this paper, we consider the exponential-type weights wρ x |x| exp −Q x , ρ > −1/2, x ∈ R, and the orthonormal polynomials pn w2 ρ;x of degree n with respect to wρ x . So, we obtain a certain differential equation of higher order with respect to pn w2 ρ;x and we estimate the higher-order derivatives of pn w2 ρ;x and the coefficients o...

Freud's Conjecture for Exponential Weights

exists. He expressed the value that the limit should take in terms of gamma functions, and proved his conjecture for a = 2,4,6. Recently, Al. Magnus [8] proved the conjecture for p > —1 and a a positive even integer, and subsequently [9] for weights of the form exp(—P(x)), where P(x) is a polynomial of even degree with positive leading coefficient. Maté, Nevai, and Zaslavsky [11] have sharpened...

Orthonormal Expansions of Invariant Densities for Expanding Maps

Asymptotic Expansions of Ratios of Coefficients of Orthogonal Polynomials with Exponential Weights

Let p,(x) = ynxn+ . . . denote the nth polynomial orthonormal with respect to the weight exp(-x8//3) where /3 > 0 is an even integer. G. Freud conjectured and Al. Magnus proved that, writing a, = y,,/y,, the expression ~ , n ' / ~ has a limit as n + w. It is shown that this expression has an asymptotic expansion in terms of negative even powers of n . In the course of this, a combinatorial enum...