An equivalent condition for q-holonomicity

TL;DR

This paper establishes a new characterization of q-holonomic sequences based on the elimination property, and demonstrates their closure properties and applications to link invariants in 3-manifolds.

Contribution

It provides an if-and-only-if condition for q-holonomicity and proves that Jones-style sequences and Reshetikhin-Turaev invariants are q-holonomic.

Findings

q-holonomic sequences satisfy the elimination property

Closure properties for q-holonomic sequences are confirmed

Jones-style sequences and invariants are q-holonomic

Abstract

We show that a sequence is q-holonomic if and only if it satisfies the elimination property for any subset of variables. The same result also holds for holonomic sequences. As an application, we prove several conjectured closure properties for q-holonomic sequences. We also prove that Jones-style sequences for links in any closed $3$-manifold are q-holonomic, which in turn implies that the Reshetikhin-Turaev invariants are q-holonomic in the colors.