Linear recurrence sequences and twisted binary forms

TL;DR

This paper investigates sequences derived from twisted binary forms, showing they are linear recurrence sequences and exploring their properties in the context of algebraic forms.

Contribution

It introduces a new class of sequences from binary forms twisted by complex numbers and proves they follow linear recurrence relations.

Findings

Sequences are linear recurrence sequences.

Explicit formulas for the recurrence relations.

Connections to algebraic forms and number theory.

Abstract

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb Z}$, which we write as $$ X^d-U_1(a)X^{d-1}Y+\cdots+(-1)^{d-1} U_{d-1}(a) XY^{d-1}+(-1)^d U_d(a) Y^d.$$ In this paper we study these sequences $\bigl(U_h(a)\bigr)_{a\in {\mathbb Z}}$ which turn out to be linear recurrence sequences.