  Laguerre and Laguerre-type polynomials are orthogonal polynomials on the
interval $[0,\infty)$ with respect to a weight function of the form $w(x) =
x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The
classical Laguerre polynomials correspond to $Q(x)=x$. The computation of
higher-order terms of the asymptotic expansions of these polynomials for large
degree becomes quite complicated, and a full description seems to be lacking in
literature. However, this information is implicitly available in the work of
Vanlessen, based on a non-linear steepest descent analysis of an associated
so-called Riemann--Hilbert problem. We will extend this work and show how to
efficiently compute an arbitrary number of higher-order terms in the asymptotic
expansions of Laguerre and Laguerre-type polynomials. This effort is similar to
the case of Jacobi and Jacobi-type polynomials in a previous paper. We supply
an implementation with explicit expansions in four different regions of the
complex plane. These expansions can also be extended to Hermite-type weights of
the form $\exp(-\sum_{k=0}^m q_k x^{2k})$ on $(-\infty,\infty)$, and to general
non-polynomial functions $Q(x)$ using contour integrals. The expansions may be
used, e.g., to compute Gauss-Laguerre quadrature rules in a lower computational
complexity than based on the recurrence relation, and with improved accuracy
for large degree. They are also of interest in random matrix theory.
