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To represent the evolution of nucleic acid and protein sequence, we expressthe parallel and Eigen models for molecular evolution in terms of a functionalintegral 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 interms of a maximum principle. Euler's theorem for homogeneous functions is usedto derive this `thermodynamic' formulation of evolution. The general result forthe 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 mutationscheme. Examples are presented for the $h=4$ and $h=20$ cases. We derive themaximum principle for the Eigen model for general $h$. The general result forthe Eigen model reduces to a known result for $h=2$. Examples are presented forthe nucleic acid $h=4$ and the amino acid $h=20$ alphabet. An error catastrophephase transition occurs in these models, and the order of the phase transitionchanges from second to first order for smooth fitness functions when thealphabet size is increased beyond two letters to the generic case. As examples,we analyze the general analytic solution for sharp peak, linear, quadratic, andquartic fitness functions.

Solution of the Crow-Kimura and Eigen models for alphabets of arbitrary size by Schwinger spin coherent states