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Volume 13
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Article

Distance Fibonacci Polynomials—Part II

by
Urszula Bednarz
* and
Małgorzata Wołowiec-Musiał
The Faculty of Mathematics and Applied Physics, Rzeszów University of Technology, Aleja Powstańców Warszawy 12, 35-959 Rzeszów, Poland
*
Author to whom correspondence should be addressed.
Symmetry 2021, 13(9), 1723; https://doi.org/10.3390/sym13091723
Submission received: 29 August 2021 / Revised: 12 September 2021 / Accepted: 14 September 2021 / Published: 17 September 2021
(This article belongs to the Special Issue Special Functions and Polynomials)
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Abstract

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials.
Keywords: generalized Fibonacci polynomials; generalized Lucas polynomials; Pascal’s triangle; generating function; matrix generator generalized Fibonacci polynomials; generalized Lucas polynomials; Pascal’s triangle; generating function; matrix generator

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MDPI and ACS Style

Bednarz, U.; Wołowiec-Musiał, M. Distance Fibonacci Polynomials—Part II. Symmetry 2021, 13, 1723. https://doi.org/10.3390/sym13091723

AMA Style

Bednarz U, Wołowiec-Musiał M. Distance Fibonacci Polynomials—Part II. Symmetry. 2021; 13(9):1723. https://doi.org/10.3390/sym13091723

Chicago/Turabian Style

Bednarz, Urszula, and Małgorzata Wołowiec-Musiał. 2021. "Distance Fibonacci Polynomials—Part II" Symmetry 13, no. 9: 1723. https://doi.org/10.3390/sym13091723

APA Style

Bednarz, U., & Wołowiec-Musiał, M. (2021). Distance Fibonacci Polynomials—Part II. Symmetry, 13(9), 1723. https://doi.org/10.3390/sym13091723

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