Orthogonal Polynomials With a Semi-Classical Weight and Their Recurrence Coefficients

Abstract

Focusing on the weight function ω(x, t) = x α e − 1 3 x 3 +tx , x ∈ [0, ∞), α > −1, t > 0, we state its asymptotic orthogonal polynomials. Through Toda evolution, differential equations of α n (t) and β n (t) have been worked. Consequently, we also talk about the approximate value of α n (t). Basing on the asymptotic value of α n (t), the asymptotic of second order differential equation of P n (z) and expansion of the logarithmic form of Hankel determinant are confirmed.

Generalised Airy Polynomials

“…In §3. 4 we consider the recurrence coefficients of polynomials orthogonal with respect to (1.3) and correct some of the results in Wang et al [33]. Properties of generalised sextic Freud polynomials and their zeros are considered in §4.…”

“…and the discrete system see equations (34), ( 35), (37) and (38) in [33]. Theorems 3.12 and 3.14 above show that their claim is not correct.…”

“…where C is a contour in the complex plane, as being for x ∈ [0, ∞); see equation ( 2) in [33]. There are other reasons to illustrate that some results in [33] are not correct. Eliminating β n in (3.32) gives…”

Generalised Airy polynomials

“…Theorems 3.12 and 3.14 above show that their claim is not correct. Wang et al [33] misquote the results of Magnus [22], who considered the weight…”

“…where C is a contour in the complex plane, as being for x ∈ [0, ∞); see equation (2) in [33]. There are other reasons to illustrate that some results in [33] are not correct. Eliminating β n in (3.35) gives…”