Automatic Generation of Convolution Identities for C-finite sequences

TL;DR

This paper introduces a rapid, experimental mathematics-based method for automatically generating convolution identities for all C-finite sequences, expanding beyond Fibonacci and Tribonacci numbers to any sequence satisfying linear recurrences.

Contribution

It presents a novel, efficient approach using the C-finite ansatz to derive convolution identities for all C-finite sequences, surpassing previous complex analysis methods.

Findings

Method can derive identities in seconds

Applicable to all C-finite sequences

Handles pairs of sequences as well

Abstract

In a recent insightful article, Helmut Prodinger uses sophisticated complex analysis, with residues, to derive convolution identities for Fibonacci, Tribonacci, and k-bonacci numbers. Here we use a naive, "experimental mathematics" (yet fully rogorous!) approach, using the C-finite ansatz, that can derive such identities in a few seconds, but not just for the above-mentioned sequences, but for every C-finite sequence (i.e. a sequence satisfying a linear recurrence with constant coefficients), and even for a pair of these.