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{\bf Paul Monsky }
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{\bf Generating Functions Attached to some Infinite Matrices}
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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.
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