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.

A weighted extension of Fibonacci numbers

“…We work in the context of elliptic combinatorics. This field has been developing rapidly, due to efforts by Schlosser, Yoo and others [3,4,12,13,14,15,16,17,18,19,20]. Our elliptic extension makes a small change in the definition of elliptic Fibonacci numbers by Schlosser and Yoo [18].…”