## 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) |
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Pages (from-to) | 39-104 |

Number of pages | 66 |

Journal | Acta Applicandae Mathematicae |

Volume | 100 |

Issue number | 1 |

DOIs | |

State | 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