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

Journal Article (Journal Article)

Let Λ ℝ denote the linear space over ℝ spanned by zk k ℤ. Define the (real) inner product Ċ,Ċ L : Λ ℝ× Λ ℝ ℝ, (f,g) ∫ℝ 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]. © 2007 Springer.

### Cited Authors

• McLaughlin, KTR; Vartanian, AH; Zhou, X

### Published Date

• March 1, 2008

• 27 / 2

• 149 - 202

• 1432-0940

• 0176-4276

### Digital Object Identifier (DOI)

• 10.1007/s00365-007-0675-z

• Scopus