  In a recent work difference equations (Laguerre-Freud equations) for the
bi-orthogonal polynomials and related quantities corresponding to the weight on
the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{\rho_j} $ were derived.Here it
is shown that in the case $ m=3 $ these difference equations, when applied to
the calculation of the underlying U(N) average, reduce to a coupled system
identifiable with that obtained by Adler and van Moerbeke using methods of the
Toeplitz lattice and Virasoro constraints. Moreover it is shown that this
coupled system can be reduced to yield the discrete fifth Painlev\'e equation
\dPV as it occurs in the theory of the sixth Painlev\'e system. Methods based
on affine Weyl group symmetries of B\"acklund transformations have previously
yielded the \dPV equation but with different parameters for the same problem.
We find the explicit mapping between the two forms. Applications of our results
are made to give recurrences for the gap probabilities and moments in the
circular unitary ensemble of random matrices, and to the diagonal spin-spin
correlation function of the square lattice Ising model.
