Elliptic Solutions of Dynamical Lucas Sequences.

Michael J Schlosser,Meesue Yoo

doi:10.3390/e23020183

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.

Highlights

In a series of papers, Lucas [1,2,3] studied the generalized Fibonacci polynomials hni which depend on two commuting variables P, Q and are defined by h0i = 0, h1i = 1, and Elliptic Solutions of Dynamical Lucas h n i = P h n − 1i + Q h n − 2i, Sequences
We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them
We show that a nice solution for this system is given in terms of elliptic numbers

Summary

Introduction

In a series of papers, Lucas [1,2,3] studied the generalized Fibonacci polynomials hni which depend on two commuting variables P, Q and are defined by h0i = 0, h1i = 1, and Elliptic Solutions of Dynamical Lucas h n i = P h n − 1i + Q h n − 2i, Sequences. We extend the sequence of these polynomials to negative indices and recover a formula by Cigler (Section 3 in [7]) for the negatively indexed non-commutative Fibonacci polynomials in terms of the non-negatively indexed ones. For elliptic-commuting variables the non-commutative Fibonacci polynomials become, what we shall call, noncommutative elliptic Fibonacci polynomials In this case after normal ordering of the ellipticcommuting variables fully factorized elliptic binomial coefficients appear in the expansion of the normalized monomials.

Elliptic Solution of a Level-Dependent Lucas System

Noncommutative Weight-Dependent Binomial Theorem

Elliptic Weights

Noncommutative Fibonacci Polynomials

Noncommutative Weight-Dependent Fibonacci Polynomials

Noncommutative Elliptic Fibonacci Polynomials