Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Discrete Painlev\'e equations, Orthogonal Polynomials on the Unit Circle
  and N-recurrences for averages over U(N) -- \PIIIa and \PV $\tau$-functions   In this work we show that the $ N\times N $ Toeplitz determinants with the
symbols $ z^{\mu}\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $
(1+z)^{\mu}(1+1/z)^{\nu}\exp(tz) $ -- known $\tau$-functions for the \PIIIa and
\PV systems -- are characterised by nonlinear recurrences for the reflection
coefficients of the corresponding orthogonal polynomial system on the unit
circle. It is shown that these recurrences are entirely equivalent to the
discrete Painlev\'e equations associated with the degenerations of the rational
surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlev\'e {\rm II}) and $
D^{(1)}_{5} \to E^{(1)}_{6} $ (discrete Painlev\'e {\rm IV}) respectively
through the algebraic methodology based upon of the affine Weyl group symmetry
of the Painlev\'e system, originally due to Okamoto. In addition it is shown
that the difference equations derived by methods based upon the Toeplitz
lattice and Virasoro constraints, when reduced in order by exact summation, are
equivalent to our recurrences. Expressions in terms of generalised
hypergeometric functions $ {{}^{\vphantom{(1)}}_0}F^{(1)}_1,
{{}^{\vphantom{(1)}}_1}F^{(1)}_1 $ are given for the reflection coefficients
respectively.

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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc