@article {IOPORT.05903052,
    author       = {Garoufalidis, Stavros},
    title        = {The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial.},
    year         = {2011},
    journal      = {The Electronic Journal of Combinatorics [electronic only]},
    volume       = {18},
    number       = {2},
    issn         = {1077-8926},
    pages        = {Research Paper P4, 23 p., electronic only},
    publisher    = {Prof. Andr\'e K\"undgen, Deptartment of Mathematics, California State University San Marcos, San Marcos, CA},
    abstract     = {Summary: 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.},
    identifier   = {05903052},
}