Elliptic solutions of dynamical Lucas sequences

TL;DR

This paper explores elliptic solutions for two types of dynamical Lucas sequences, including level-dependent and non-commutative versions, revealing new properties and identities involving elliptic numbers and polynomials.

Contribution

It introduces elliptic solutions to dynamical Lucas sequences, including non-commutative elliptic Fibonacci polynomials, and derives their explicit expansions and identities.

Findings

Elliptic numbers solve the level-dependent Lucas sequence system.

Non-commutative elliptic Fibonacci polynomials are explicitly expanded.

Derived a non-commutative elliptic Euler--Cassini identity.

Abstract

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler--Cassini identity.