### Abstract

Let Λ ^{ℝ} denote the linear space over ℝ spanned by z^{k} 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(x^{2}} + 1)) = + and (iii) limx 0(V(x)/ln (x^{2}} + 1)) = +. Orthogonalisation of the (ordered) base 1,z^{-1},z,z^{-2}z^{2}},z ^{-k}},z^{k} with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z)_{m=0}: φ_{2n}(z) = ξ^{(2n)}z^{-n} + + ξ^{(2n)}_{n}z^{n} ξ^{(2n)}_{n} > 0, and φ^{{2n+1}}(z) = ξ^{(2n+1)} _{-n-1}z^{-n-1} + + ξ^{(2n+1)n}z ^{n} ξ^{(2n+1)}_{-n-1} > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π_{2n}(z) := (ξ^{(2n)}_{n}^{-1}} φ_{2n}(z) and π^{{2n+1}}(z) := (ξ^{(2n+1)}_{-n-1}^{-1} φ_{2n+1}(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π_{2n+1}(z) (in the entire complex plane), ξ^{(2n+1)}_{-n-1}, and φ_{2n+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].

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

Pages (from-to) | 149-202 |

Number of pages | 54 |

Journal | Constructive Approximation |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2008 |

### Fingerprint

### Keywords

- Asymptotics
- Equilibrium measures
- Hankel determinants
- Laurent polynomials
- Laurent-Jacobi matrices
- Riemann-Hilbert problems

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Constructive Approximation*,

*27*(2), 149-202. https://doi.org/10.1007/s00365-007-0675-z

**Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights.** / Mclaughlin, Kenneth D T; Vartanian, A. H.; Zhou, X.

Research output: Contribution to journal › Article

*Constructive Approximation*, vol. 27, no. 2, pp. 149-202. https://doi.org/10.1007/s00365-007-0675-z

}

TY - JOUR

T1 - Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights

AU - Mclaughlin, Kenneth D T

AU - Vartanian, A. H.

AU - Zhou, X.

PY - 2008/3

Y1 - 2008/3

N2 - 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].

AB - 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].

KW - Asymptotics

KW - Equilibrium measures

KW - Hankel determinants

KW - Laurent polynomials

KW - Laurent-Jacobi matrices

KW - Riemann-Hilbert problems

UR - http://www.scopus.com/inward/record.url?scp=34548540038&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548540038&partnerID=8YFLogxK

U2 - 10.1007/s00365-007-0675-z

DO - 10.1007/s00365-007-0675-z

M3 - Article

VL - 27

SP - 149

EP - 202

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 2

ER -