# Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights

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

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

## Abstract

Let Λ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λ×Λ→ℝ, (f,g)∫}f(s)g(s)exp (-{N}V(s)){d}s, N ∈, where V satisfies: (i) V is real analytic on ℝ/{0}; (ii) lim∈ | x |→∞(V(x)/ln∈(x 2+1))=+∞; and (iii) lim∈ | x |→0(V(x)/ln∈(x -2+1))= +∞. Orthogonalisation of the (ordered) base with respect to 〈 .,.〉 L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ 2n (z)= k=-n n ξ k (2n) z k , ξ n (2n) >0, and φ 2n+1(z)= k=-n-1 n ξ k (2n+1) z k , ξ -n-1 (2n+1) >0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ 2n (z)=c 2n # φ 2n-2(z)+b 2n # φ 2n-1(z)+a 2n # φ 2n (z)+b 2n+1 # φ 2n+1(z)+c 2n+2 # φ 2n+2(z) and z φ 2n+1(z)=b 2n+1 # φ 2n (z)+a 2n+1 # φ 2n+1(z)+b 2n+2 # φ 2n+2(z), where c 0 # =b 0 # =0, and c 2k # >0, k ∈, and z -1 φ 2n+1(z)=γ 2n+1 # φ 2n-1(z)+β 2n+1 # φ 2n (z)+α 2n+1 # φ 2n+1(z)+β 2n+2 # φ 2n+2(z)+γ 2n+3 # φ 2n+3(z) and z -1 φ 2n (z)=β 2n # φ 2n-1(z)+α 2n # φ 2n (z)+β 2n+1 # φ 2n+1(z), where β 0 #1 # =0, β 1 # >0, and γ 2l+1 # >0, l ∈. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence k= ∫, and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295-368, ) and further developed in (Commun. Pure Appl. Math. 48(3):277-337, ) and (Int. Math. Res. Not. 6:285-299, ).

Original language English (US) 39-104 66 Acta Applicandae Mathematicae 100 1 https://doi.org/10.1007/s10440-007-9176-0 Published - Jan 2008

## Keywords

• Asymptotics
• Equilibrium measures
• Hankel determinants
• Laurent-Jacobi matrices
• Orthogonal Laurent polynomials
• Recurrence relations
• Riemann-Hilbert problems
• Variational problems

## ASJC Scopus subject areas

• Applied Mathematics

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