### Abstract

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e^{-Q(x)}dx on the real line, where Q(x) = Σ^{2m}_{k=0}q_{kxk}, q_{2m} > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials.

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

Pages (from-to) | 1491-1552 |

Number of pages | 62 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 52 |

Issue number | 12 |

State | Published - Dec 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*52*(12), 1491-1552.

**Strong asymptotics of orthogonal polynomials with respect to exponential weights.** / Deift, P.; Kriecherbauer, T.; Mclaughlin, Kenneth D T; Venakides, S.; Zhou, X.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 52, no. 12, pp. 1491-1552.

}

TY - JOUR

T1 - Strong asymptotics of orthogonal polynomials with respect to exponential weights

AU - Deift, P.

AU - Kriecherbauer, T.

AU - Mclaughlin, Kenneth D T

AU - Venakides, S.

AU - Zhou, X.

PY - 1999/12

Y1 - 1999/12

N2 - We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e-Q(x)dx on the real line, where Q(x) = Σ2mk=0qkxk, q2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials.

AB - We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e-Q(x)dx on the real line, where Q(x) = Σ2mk=0qkxk, q2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials.

UR - http://www.scopus.com/inward/record.url?scp=0033459230&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033459230&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033459230

VL - 52

SP - 1491

EP - 1552

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 12

ER -