Asymptotics of determinants of Hankel matrices via non-linear difference equations | |
Estelle L. Basor1; Yang Chen2; Nazmus S. Haq3 | |
2015-10 | |
Source Publication | Journal of Approximation Theory |
ISSN | 0021-9045 |
Volume | 198Pages: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], k^{2}∈(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 k^{2}, 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(k^{2}), n=1, 2, ...; and p1(n, k^{2}), the coefficients of x^{n-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. |
Keyword | Asymptotic Expansions Elliptic Orthogonal Polynomials Hankel Determinants Non-linear Difference Equations Painlevé Equations Random Matrix Theory |
DOI | 10.1016/j.jat.2015.05.002 |
URL | View the original |
Indexed By | SCI |
Language | 英语 |
WOS Research Area | Mathematics |
WOS Subject | Mathematics |
WOS ID | WOS:000359876200005 |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | DEPARTMENT OF MATHEMATICS |
Corresponding Author | Nazmus S. Haq |
Affiliation | 1.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. |
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