Michael J. Schlosser and Meesue Yoo
Elliptic solutions of dynamical Lucas sequences
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
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