Elliptic and q-Analogs of the Fibonomial Numbers

Lopez, Cesar Ceballos; Bergeron, Nantel; Küstner, Josef

doi:10.3842/sigma.2020.076

“…We would like to point out that the results in the current paper do not appear to directly contain the elliptic Fibonacci numbers which were introduced in [12] nor those (of a simpler type) which were introduced in [1]. While we believe that there is a connection of our non-commutative elliptic Fibonacci polynomials considered in Section 4 of this paper with our earlier elliptic Fibonacci numbers in [12], the connection is not yet entirely clear and requires further investigations.…”

Section: Introductionmentioning

confidence: 74%

Elliptic solutions of dynamical Lucas sequences

Schlosser,

Yoo

2020

Preprint

0

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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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“…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].…”

Section: Introductionmentioning

confidence: 99%

A weighted extension of Fibonacci numbers

Bhatnagar¹,

Kumari²,

Schlosser³

2023

Preprint

0

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We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the q-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the q case, by Garrett. Some identities are proved by telescoping. the principal transcendental functions, Reprint of the fourth (1927) edition.

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“…We would like to point out that the results in the current paper do not appear to directly contain the elliptic Fibonacci numbers which were introduced in [ 8 ] nor those (of a simpler type) which were introduced in [ 9 ]. While we believe that there is a connection of our non-commutative elliptic Fibonacci polynomials considered in Section 4 of this paper with our earlier elliptic Fibonacci numbers in [ 8 ], the connection is not yet entirely clear and requires further investigations.…”

Section: Introductionmentioning

confidence: 99%

Elliptic Solutions of Dynamical Lucas Sequences

Schlosser

¹

,

Yoo

²

2021

Entropy

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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.

show abstract

Elliptic and q-Analogs of the Fibonomial Numbers

Cited by 4 publications

References 21 publications

Elliptic solutions of dynamical Lucas sequences

Elliptic solutions of dynamical Lucas sequences

A weighted extension of Fibonacci numbers

Elliptic Solutions of Dynamical Lucas Sequences

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