# Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

## TL;DR

This paper extends methods for bounding the denominators and degrees of polynomial solutions to systems of higher order q-recurrence equations with rational coefficients, without relying on uncoupling or reduction techniques.

## Contribution

It generalizes scalar case techniques to systems of q-recurrences, providing direct methods for bounds without uncoupling or reduction.

## Key findings

- Provides bounds on the maximal power of t in denominators of solutions
- Establishes degree bounds for polynomial solutions
- Offers a direct approach that avoids uncoupling or reduction

## Abstract

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.