  To represent the evolution of nucleic acid and protein sequence, we express
the parallel and Eigen models for molecular evolution in terms of a functional
integral representation with an $h$-letter alphabet, lifting the two-state,
purine/pyrimidine assumption often made in quasi-species theory. For arbitrary
$h$ and a general mutation scheme, we obtain the solution of this model in
terms of a maximum principle. Euler's theorem for homogeneous functions is used
to derive this `thermodynamic' formulation of evolution. The general result for
the parallel model reduces to known results for the purine/pyrimidine $h=2$
alphabet and the nucleic acid $h=4$ alphabet for the Kimura 3 ST mutation
scheme. Examples are presented for the $h=4$ and $h=20$ cases. We derive the
maximum principle for the Eigen model for general $h$. The general result for
the Eigen model reduces to a known result for $h=2$. Examples are presented for
the nucleic acid $h=4$ and the amino acid $h=20$ alphabet. An error catastrophe
phase transition occurs in these models, and the order of the phase transition
changes from second to first order for smooth fitness functions when the
alphabet size is increased beyond two letters to the generic case. As examples,
we analyze the general analytic solution for sharp peak, linear, quadratic, and
quartic fitness functions.
