### Abstract

Let Λ^{ℝ} denote the linear space over ℝ spanned by z ^{k} , k ∈ ℤ. Define the real inner product 〈 .,.〉 _{L} : Λ^{ℝ}×Λ^{ℝ}→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, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277-337, [1995]) and (Int. Math. Res. Not. 6:285-299, [1997]).

Original language | English (US) |
---|---|

Pages (from-to) | 39-104 |

Number of pages | 66 |

Journal | Acta Applicandae Mathematicae |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2008 |

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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

*Acta Applicandae Mathematicae*,

*100*(1), 39-104. https://doi.org/10.1007/s10440-007-9176-0

**Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights.** / Mclaughlin, Kenneth D T; Vartanian, A. H.; Zhou, X.

Research output: Contribution to journal › Article

*Acta Applicandae Mathematicae*, vol. 100, no. 1, pp. 39-104. https://doi.org/10.1007/s10440-007-9176-0

}

TY - JOUR

T1 - Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights

AU - Mclaughlin, Kenneth D T

AU - Vartanian, A. H.

AU - Zhou, X.

PY - 2008/1

Y1 - 2008/1

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, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277-337, [1995]) and (Int. Math. Res. Not. 6:285-299, [1997]).

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, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277-337, [1995]) and (Int. Math. Res. Not. 6:285-299, [1997]).

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

UR - http://www.scopus.com/inward/record.url?scp=37549070766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37549070766&partnerID=8YFLogxK

U2 - 10.1007/s10440-007-9176-0

DO - 10.1007/s10440-007-9176-0

M3 - Article

AN - SCOPUS:37549070766

VL - 100

SP - 39

EP - 104

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

SN - 0167-8019

IS - 1

ER -