Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $τ$-functions
| dc.creator | Forrester, P. J. | |
| dc.creator | Witte, N. S. | |
| dc.date | 2003-08-28 | |
| dc.date.accessioned | 2026-07-07T04:30:30Z | |
| dc.date.available | 2026-07-07T04:30:30Z | |
| dc.description | The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting of $ m \in \mathbb{Z}_{> 0} $ singularities, difference equations with respect to the orthogonal polynomial degree $ n $ (Laguerre-Freud equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system. It is shown in the simplest non-trivial case of $ m=3 $ that quite generally and simply the difference equations are equivalent to the discrete Painlevé equation associated with the degeneration of the rational surface $ D^{(1)}_4 \to D^{(1)}_5 $ and no other. In a three way comparison with other methods employed on this problem - the Toeplitz lattice and Virasoro constraints, the isomonodromic deformation of $ 2\times 2 $ linear Fuchsian differential equations, and the algebraic approach based upon the affine Weyl group symmetry - it is shown all are entirely equivalent, when reduced in order by exact summation, to the above discrete Painlevé equation through explicit transformation formulae. The fundamental matrix integrals over the unitary group $ U(N) $ arising in the theory are given by the generalised hypergeometric function $ {{}^{\vphantom{(1)}}_2}F^{(1)}_1 $. | |
| dc.description | 46 pages, AMSLatex | |
| dc.identifier | https://arxiv.org/abs/math-ph/0308036 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0308036 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57484 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 05E35;39A05;37F10;33C45;34M55 | |
| dc.title | Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $τ$-functions | |
| dc.type | text |