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

Abstract

A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. 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 Dmodules 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 aWKB 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.