Fast Computation of the $N$-th Term of a $q$-Holonomic Sequence and Applications

TL;DR

This paper introduces $q$-analogues of Strassen's and Chudnovsky's algorithms, enabling fast computation of the $N$-th term of $q$-holonomic sequences with quasi-linear complexity, and explores their applications.

Contribution

The paper develops simpler $q$-analogues of classical algorithms for holonomic sequences, achieving quasi-linear complexity and broadening computational tools for $q$-series.

Findings

Algorithms compute $q$-factorials and $q$-holonomic sequence terms in quasi-linear time.

The methods are simpler than their classical counterparts.

Applications include accelerating polynomial and rational $q$-differential equation solving.

Abstract

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen's algorithm to the computation of the $q$-factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$-th term of any $q$-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.