  The Toeplitz determinants (of increasing size) associated with the symbols
$exp{t(z+z^{-1})}$ or $(1-{\xi}{z})^{\alpha} (1-{\xi}{z^{-1}})^{\beta}$ satisfy
recursion relations, thus expressing all the Toeplitz determinants as a
rational function of the first few determinants. A. Borodin found these
relations using Riemann-Hilbert methods.
  The nature of Borodin's relations pointed towards the Toeplitz lattice and
its Virasoro algebra, as developed by the authors. In this paper, we take the
Toeplitz and Virasoro approach for a fairly large class of symbols, leading to
a systematic and simple way of generating such recursion relations.
  The latter are very naturally expressed in terms of the $L$-matrices
appearing in the Lax pair for the Toeplitz lattice equations. As a surprise, we
find, compared to Borodin's, a different set of relations, except for the
3-step relations associated with the symbol $ e^{t(z+z^{-1})}$. Moreover, these
recursion relations define an invariant manifold for the Toeplitz lattice. This
leads to a "discrete Painlev\'e property" (singularity confinement) for the
rational recursion relations, as a consequence of the classical ``continuous
Painlev\'e property" for the Toeplitz lattice.
