Elliptic solutions of dynamical Lucas sequences
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.
Keywords
Cite
@article{arxiv.2012.15794,
title = {Elliptic solutions of dynamical Lucas sequences},
author = {Michael J. Schlosser and Meesue Yoo},
journal= {arXiv preprint arXiv:2012.15794},
year = {2021}
}
Comments
16 pages; minor changes, Eq. (3.10) corrected; dedicated to Johann Cigler on the occasion of his 84th birthday