Generating functions attached to some infinite matrices (Q625364)

Summary: 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'' \(G=\sum a_{1,1}(V^n)z^n\). \textit{I. Gessel} [J. Comb. Theory, Ser. A 28, 321--337 (1980)] has shown that \(G\) is algebraic over \(F(z)\). We extend his result, allowing \(v_{ij}\) 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.