  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))^{\rho_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
Painlev\'e 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 Painlev\'e 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 $.
