| Status | 已發表Published |
| Journal of Approximation Theory | |
Basor, E.; Chen, Y. ; Haq, N.
| |
| 2015-10-01 | |
| Source Publication | Journal of Approximation Theory
 |
| ISSN | 0021-9045 |
| Pages | 63-110 |
| Abstract | E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight [x(x − α)(x − β)]−1 2 , 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) −1 2 , 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 0 < k2 < 1 is the τ function of a particular Painlev´e 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. |
| Keyword | Random Matrices Orthogonal Polynomials Painleve equations |
| Language | 英語English |
| The Source to Article | PB_Publication |
| PUB ID | 15301 |
| Document Type | Journal article |
| Collection | DEPARTMENT OF MATHEMATICS |
| Corresponding Author | Chen, Y. |
| Recommended Citation GB/T 7714 | Basor, E.,Chen, Y.,Haq, N.. Journal of Approximation Theory[J]. Journal of Approximation Theory, 2015, 63-110. |
| APA | Basor, E.., Chen, Y.., & Haq, N. (2015). Journal of Approximation Theory. Journal of Approximation Theory, 63-110. |
| MLA | Basor, E.,et al."Journal of Approximation Theory".Journal of Approximation Theory (2015):63-110. |
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