UM  > 科技學院  > 數學系
Asymptotics of determinants of Hankel matrices via non-linear difference equations
Estelle L. Basor1; Yang Chen2; Nazmus S. Haq3
2015-10
Source PublicationJournal of Approximation Theory
ISSN0021-9045
Volume198Pages:63-110
Abstract

E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight [x(x-α)(x-β)]-12, x∈[0, α], 0<α<β. A related system was studied by C. J. Rees in 1945, associated with the weight [(1-x2)(1-k2x2)]-12, x∈[-1, 1], k2∈(0, 1). These are also known as elliptic orthogonal polynomials, since the moments of the weights may be expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter k2, where 02<1 is the τ function of a particular Painlevé VI, the special cases of which are related to enumerative problems arising from string theory. We show that the recurrence coefficients, denoted by βn(k2), n=1, 2, ...; and p1(n, k2), the coefficients of xn-2 of the monic polynomials orthogonal with respect to a generalized version of the weight studied by Rees, (1-x2)α(1-k2x2)β,x∈[-1,1],α>-1,β∈R, satisfy second order non-linear difference equations. The large n expansion based on the difference equations when combined with known asymptotics of the leading terms of the associated Hankel determinant yields a complete asymptotic expansion of the Hankel determinant. The Painlevé equation is also discussed as well as the generalization of the linear second order differential equation found by Rees.

KeywordAsymptotic Expansions Elliptic Orthogonal Polynomials Hankel Determinants Non-linear Difference Equations Painlevé Equations Random Matrix Theory
DOI10.1016/j.jat.2015.05.002
URLView the original
Indexed BySCI
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000359876200005
Fulltext Access
Citation statistics
Cited Times [WOS]:4   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorNazmus S. Haq
Affiliation1.American Institute of Mathematics, 360 Portage Avenue, Palo Alto, CA 94306-2244, USA
2.Faculty of Science and Technology, Department of Mathematics, University of Macau, Av. Padre Tomas Pereira, Taipa ´ Macau, China
3.Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
Recommended Citation
GB/T 7714
Estelle L. Basor,Yang Chen,Nazmus S. Haq. Asymptotics of determinants of Hankel matrices via non-linear difference equations[J]. Journal of Approximation Theory,2015,198:63-110.
APA Estelle L. Basor,Yang Chen,&Nazmus S. Haq.(2015).Asymptotics of determinants of Hankel matrices via non-linear difference equations.Journal of Approximation Theory,198,63-110.
MLA Estelle L. Basor,et al."Asymptotics of determinants of Hankel matrices via non-linear difference equations".Journal of Approximation Theory 198(2015):63-110.
Related Services
Recommend this item
Bookmark
Usage statistics
Export to Endnote
Google Scholar
Similar articles in Google Scholar
[Estelle L. Basor]'s Articles
[Yang Chen]'s Articles
[Nazmus S. Haq]'s Articles
Baidu academic
Similar articles in Baidu academic
[Estelle L. Basor]'s Articles
[Yang Chen]'s Articles
[Nazmus S. Haq]'s Articles
Bing Scholar
Similar articles in Bing Scholar
[Estelle L. Basor]'s Articles
[Yang Chen]'s Articles
[Nazmus S. Haq]'s Articles
Terms of Use
No data!
Social Bookmark/Share
All comments (0)
No comment.
 

Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.