On Some Families of Certain Divisible Polynomials and Their Zeta Functions

Abstract

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 similar divisible polynomials. Among them, three families are dealt with in this paper and their properties are investigated: they are analogues of the Mallows--Sloane bound, the extremal property, the Riemann hypothesis analogue, 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.