Geodesic
The degree of a \(q\)-holonomic sequence is a quadratic quasi-polynomial
The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2
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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, and that the leading term satisfies a linear recursion relation with constant coefficients. 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 $q=0$. We then use the Skolem-Mahler-Lech theorem to study the vanishing of their leading term. Unlike the case of $q=1$, there are no analytic problems regarding convergence of the WKB solutions. Our proofs are constructive, and they are illustrated by an explicit example.
DOI : 10.37236/2000
Classification : 05A30
Mots-clés : degree of a \(q\)-holonomic sequence, quadratic quasi-polynomial 
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     author = {Stavros Garoufalidis},
     title = {The degree of a \(q\)-holonomic sequence is a quadratic quasi-polynomial},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {2},
     doi = {10.37236/2000},
     zbl = {1298.05042},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/2000/}
}
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Stavros Garoufalidis. The degree of a \(q\)-holonomic sequence is a quadratic quasi-polynomial. The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2. doi: 10.37236/2000

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