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2010

The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial

Garoufalidis, Stavros

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A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$.

We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial.
Our proof uses differential Galois theory (adapting proofs regarding holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory.
En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB basis of solutions of a linear $q$-difference equation at $q=0$.

The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial · Around

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Kurt Mahler,
1935
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Th. Skolem,
1935
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Robert J. Walker,
1950
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Christer Lech,
1953
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Richard P. Stanley,
1986
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A. J. van der Poorten,
1989
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Doron Zeilberger,
1990
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Jean-Paul Bézivin and Abdelbaki Boutabaa,
1992
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