  We consider polynomials that are orthogonal on $[-1,1]$ with respect to a
modified Jacobi weight $(1-x)^\alpha (1+x)^\beta h(x)$, with $\alpha,\beta>-1$
and $h$ real analytic and stricly positive on $[-1,1]$. We obtain full
asymptotic expansions for the monic and orthonormal polynomials outside the
interval $[-1,1]$, for the recurrence coefficients and for the leading
coefficients of the orthonormal polynomials. We also deduce asymptotic behavior
for the Hankel determinants. For the asymptotic analysis we use the steepest
descent technique for Riemann--Hilbert problems developed by Deift and Zhou,
and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer,
McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the
Szeg\H{o} function associated with the weight and for the local analysis around
the endpoints $\pm 1$ we use Bessel functions of appropriate order, whereas
Deift et al. use Airy functions.
