Higher $q$-Continued Fractions

TL;DR

This paper introduces a $q$-analog of higher continued fractions, generalizing previous $q$-rational numbers and providing matrix formulas for their computation, while preserving key properties.

Contribution

It defines a new $q$-analog of higher continued fractions as ratios of generating functions for $P$-partitions, extending prior work and offering generalized matrix formulas.

Findings

Provides matrix formulas for the $q$-higher continued fractions.

Shows that properties of $q$-rationals extend to the higher versions.

Generalizes previous results in the case $q=1$.

Abstract

We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and Ovsienko. They are defined as ratios of generating functions for $P$-partitions on certain posets. We give matrix formulas for computing them, which generalize previous results in the $q=1$ case. We also show that certain properties enjoyed by the $q$-rationals are also satisfied by our higher versions.