Elliptic Solutions of Dynamical Lucas Sequences |
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Authors: | Michael J. Schlosser Meesue Yoo |
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Affiliation: | 1.Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria;2.Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea; |
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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. |
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Keywords: | Lucas sequences theta functions elliptic numbers non-commutative Fibonacci polynomials |
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