Decomposable forms generated by linear recurrences

TL;DR

This paper explores the connection between linear recurrences and decomposable forms, providing identities, factorizations, and solutions for associated Diophantine equations, with explicit examples for low orders.

Contribution

It establishes a broad general identity linking linear recurrences to decomposable forms and offers a method for their complete factorization, including explicit cases for k=2 and examples for k=3.

Findings

Identifies a general identity relating recurrences and decomposable forms

Provides a method for factorization of these forms

Shows infinite integer solutions for associated Diophantine equations

Abstract

Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broad general identity with a suitable exponential expression depending on the recurrences. This identity is a common and wide generalization of several known identities. Further, if the recurrences are integer sequences, then the diophantine equation associated to the decomposable form and the exponential term has infinitely many integer solutions generated by the terms of the recurrences. We describe a method for the complete factorization of the decomposable form. Both the form and its decomposition are explicitly given if $k=2$, and we present a typical example for $k=3$. The basic tool we use is the matrix method.