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

Kenneth D T Mclaughlin, A. H. Vartanian, X. Zhou

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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

### Fingerprint

Orthogonal Laurent Polynomials
Hankel Determinant
Exponential Weights
Orthonormal Polynomials
Laurent Polynomials
Recurrence relation
Polynomials
Roots
Odd
Coefficient
Moment Sequence
Steepest descent method
Steepest Descent Method
Orthogonalization
Real Roots
Riemann-Hilbert Problem
Scaling Limit
Linear Space
Scalar, inner or dot product
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### Keywords

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

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

In: Acta Applicandae Mathematicae, Vol. 100, No. 1, 01.2008, p. 39-104.

Research output: Contribution to journalArticle

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title = "Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights",
abstract = "Let Λℝ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λℝ×Λℝ→ℝ, (f,g)∫ℝ}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 # =γ 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= ∫ℝ, 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, ).",
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author = "Mclaughlin, {Kenneth D T} and Vartanian, {A. H.} and X. Zhou",
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T1 - Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights

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AU - Vartanian, A. H.

AU - Zhou, X.

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N2 - Let Λℝ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λℝ×Λℝ→ℝ, (f,g)∫ℝ}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 # =γ 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= ∫ℝ, 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, ).

AB - Let Λℝ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λℝ×Λℝ→ℝ, (f,g)∫ℝ}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 # =γ 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= ∫ℝ, 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, ).

KW - Asymptotics

KW - Equilibrium measures

KW - Hankel determinants

KW - Laurent-Jacobi matrices

KW - Orthogonal Laurent polynomials

KW - Recurrence relations

KW - Riemann-Hilbert problems

KW - Variational problems

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