# Elliptic Solutions of Dynamical Lucas Sequences

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## Abstract

**:**

## 1. Introduction

## 2. Elliptic Solution of a Level-Dependent Lucas System

## 3. Weight-Dependent Commutation Relations and Elliptic Weights

#### 3.1. Noncommutative Weight-Dependent Binomial Theorem

**Definition**

**1.**

**Theorem**

**1**

**.**Let $n\in {\mathbb{N}}_{0}$. Then, as an identity in ${\mathbb{C}}_{w}[x,y]$,

**Lemma**

**1**

**.**We have

**Definition**

**2.**

**Lemma**

**2.**

#### 3.2. Elliptic Weights

**Proposition**

**1.**

**Definition**

**3.**

**Definition**

**4.**

## 4. Noncommutative Fibonacci Polynomials

**Example**

**1.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

#### 4.1. Noncommutative Weight-Dependent Fibonacci Polynomials

**Proposition**

**3.**

**Proof.**

#### 4.2. Noncommutative Elliptic Fibonacci Polynomials

**Corollary**

**1.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Schlosser, M.J.; Yoo, M.
Elliptic Solutions of Dynamical Lucas Sequences. *Entropy* **2021**, *23*, 183.
https://doi.org/10.3390/e23020183

**AMA Style**

Schlosser MJ, Yoo M.
Elliptic Solutions of Dynamical Lucas Sequences. *Entropy*. 2021; 23(2):183.
https://doi.org/10.3390/e23020183

**Chicago/Turabian Style**

Schlosser, Michael J., and Meesue Yoo.
2021. "Elliptic Solutions of Dynamical Lucas Sequences" *Entropy* 23, no. 2: 183.
https://doi.org/10.3390/e23020183