# Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights

K. T.R. McLaughlin, A. H. Vartanian, X. Zhou

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

## Abstract

Let Λ denote the linear space over ℝ spanned by zk k ℤ. Define the (real) inner product Ċ,Ċ L : Λ × Λ ℝ, (f,g) ∫ f(s)g(s) exp(- N V(s)) ds, N ℕ, where V satisfies: (i) V is real analytic on ℝ 0; (ii) lim x (V(x)/ln(x2} + 1)) = + and (iii) limx 0(V(x)/ln (x2} + 1)) = +. Orthogonalisation of the (ordered) base 1,z-1,z,z-2z2},z -k},zk with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z)m=0: φ2n(z) = ξ(2n)z-n + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1) -n-1z-n-1 + + ξ(2n+1)nz n ξ(2n+1)-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n-1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)-n-1-1 φ2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)-n-1, and φ2n+1(z)(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ℝ, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].

Original language English (US) 149-202 54 Constructive Approximation 27 2 https://doi.org/10.1007/s00365-007-0675-z Published - Mar 2008

## Keywords

• Asymptotics
• Equilibrium measures
• Hankel determinants
• Laurent polynomials
• Laurent-Jacobi matrices
• Riemann-Hilbert problems

## ASJC Scopus subject areas

• Analysis
• Mathematics(all)
• Computational Mathematics