# On Eisenstein polynomials and zeta polynomials II

## TL;DR

This paper extends the analogy between Eisenstein series and Eisenstein polynomials, demonstrating that certain properties hold across multiple types, thus deepening the understanding of their structural similarities.

## Contribution

It proves that the analogous properties of Eisenstein polynomials and zeta polynomials apply to Type I, III, and IV cases, broadening previous results.

## Key findings

- Properties hold for Type I, III, and IV Eisenstein polynomials
- Deepens the analogy between Eisenstein series and polynomials
- Extends previous Type II results

## Abstract

Eisenstein polynomials, which were defined by the second author, are analogues of the concept of an Eisenstein series. The second author conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In the previous paper, the first author provided new analogous properties of Eisenstein polynomials and zeta polynomials for the Type II case. In this paper, the analogous properties of Eisenstein polynomials and zeta polynomials are shown to also hold for the Type I, Type III, and Type IV cases. These properties are finite analogies of certain properties of Eisenstein series.