#### Title

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

#### Abbreviated Journal Title

Int. Math. Res. Pap.

#### Keywords

2-POINT PADE APPROXIMANTS; HAMBURGER MOMENT PROBLEM; RIEMANN-HILBERT; APPROACH; INVERSE SCATTERING TRANSFORMS; RELATIVISTIC TODA LATTICE; STEEPEST DESCENT METHOD; RANDOM-MATRIX THEORY; RATIONAL FUNCTIONS; UNIT-CIRCLE; STIELTJES FUNCTIONS; Mathematics

#### Abstract

Let Lambda(R) denote the linear space over R spanned by z(k), k is an element of Z. Define the real inner product (with varying exponential weights) [center dot, center dot](L) : Lambda(R) x Lambda(R) -> R, (f, g) /-> integral(R) f(s)g(s) exp(-NV(s))ds, N is an element of N, where the external field V satisfies the following: (i) V is real analytic on R\{0}; (ii) lim(\x\->infinity) (V(x)/ln(x(2) + 1)) = +infinity; and (iii) lim(\x\-> 0) (V(x)/ln(x(-2) + 1)) = +infinity. Orthogonalisation of the (ordered) base {1, z(-1), z, z(-2), z(2), ..., z(-k), z(k), ...} with respect to [center dot, center dot](L) yields the even degree and odd degree orthonormal Laurent polynomials {phi(m)(z)}(m=0)(infinity): phi(2n)(z) = xi((2n))(-n)z(-n) + center dot center dot center dot + xi((2n))(n)z(n), xi((2n))(n) > 0, and phi(2n+1)(z) = xi((2n+1))(-n-1)z(-n-1) + center dot center dot center dot + xi((2n+1))(n)z(n), xi((2n+1))(-n-1) > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: pi(2n)(z) := (xi((2n))(n))(-1) phi(2n)(z) and pi(2n+1)(z) := (xi((2n+1))(-n-1))-1 phi(2n+1)(z). Asymptotics in the double-scaling limit as N, n -> infinity such that N/n = 1 + o(1) of pi(2n)(z) (in the entire complex plane), xi((2n))(n), phi(2n)(z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {c(k) = integral(R) s(k) exp(-NV(s))ds}(k is an element of Z) are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the large-n behaviour by applying the nonlinear steepest-descent method introduced by P. Deift and X. Zhou and further developed by P. Deift, S. Venakides, and X.

#### Journal Title

International Mathematics Research Papers

#### Publication Date

1-1-2006

#### Document Type

Review

#### Language

English

#### First Page

216

#### WOS Identifier

#### ISSN

1687-3017

#### Recommended Citation

"Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights" (2006). *Faculty Bibliography 2000s*. 6417.

http://stars.library.ucf.edu/facultybib2000/6417

## Comments

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