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تسجيل مستخدم جديدElliptic solutions of dynamical Lucas sequences
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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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Given two variables $s$ and $t$, the associated sequence of Lucas polynomials is defined inductively by ${0}=0$, ${1}=1$, and ${n}=s{n-1}+t{n-2}$ for $nge2$. An integer (e.g., a Catalan number) defined by an expression of the form $prod_i n_i/prod_j
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