The C-finite Ansatz Meets the Holonomic Ansatz

TL;DR

This paper introduces a method using the Almkvist-Zeilberger algorithm to automatically find and prove differential equations for generating functions of sequences derived from C-finite polynomials, enabling automatic derivation of recurrences.

Contribution

It presents a novel application of the Almkvist-Zeilberger algorithm to sequences from C-finite polynomials, with an implementation in a Maple package for automation.

Findings

Successfully derives differential equations for generating functions.

Automates the discovery of linear recurrences for sequences.

Provides a Maple package for practical use.

Abstract

We show how the continuous Almkvist-Zeilberger algorithm can be used to efficiently discover and prove differential equations satisfied by generating functions of sequences defined as integrals of powers of C-finite polynomial sequences (like the Chebyshev polynomials) from which one can automatically derive linear recurrences with polynomial coefficients for the sequences themselves. We fully implement this with a Maple package, CfiniteIntergal.txt .