On some families of divisible formal weight enumerators and their zeta functions

The formal weight enumerators were first introduced by M. Ozeki, and it was shown in the author's previous paper that there are various families of divisible formal weight enumerators. Among them, three families are dealt with in this paper and their properties are investigated: they are analogs of the Mallows-Sloane bound, the extremal property, the Riemann hypothesis, etc. In the course of the investigation, some generalizations of the theory of invariant differential operators developed by I. Duursma and T. Okuda are deduced.