Generating functions attached to some infinite matrices
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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.
@article{10_37236_492,
author = {Paul Monsky},
title = {Generating functions attached to some infinite matrices},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/492},
zbl = {1209.05264},
url = {http://geodesic.mathdoc.fr/articles/10.37236/492/}
}
Paul Monsky. Generating functions attached to some infinite matrices. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/492
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