Generating functions attached to some infinite matrices

TL;DR

This paper extends Gessel's result on generating functions of infinite matrices with banded structure, showing they are algebraic even when entries are eventually periodic, with applications to Hilbert-Kunz theory.

Contribution

It generalizes Gessel's algebraicity result to matrices with eventually periodic entries, broadening the class of matrices for which generating functions are algebraic.

Findings

The generating function G is algebraic over F(z) for matrices with eventually periodic entries.

The results have potential applications in Hilbert-Kunz theory.

Extensions include variants of the main algebraic property.

Abstract

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating function" G=\sum a_{1,1}(V^n)z^n. Ira Gessel has shown that G is algebraic over F(z). We extend his result, allowing v_{i,j} for fixed i-j to be eventually periodic in i rather than constant. This result and some variants of it that we prove will have applications to Hilbert-Kunz theory.