Distance Fibonacci Polynomials—Part II

Urszula Bednarz,Małgorzata Wołowiec-Musiał

doi:10.3390/sym13091723

Urszula Bednarz, Małgorzata Wołowiec-Musiał

Open Access

https://doi.org/10.3390/sym13091723

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Journal: Symmetry	Publication Date: Sep 17, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Rzeszów University of Technology

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.

Highlights

Fibonacci polynomials f n ( x ) are given by the recurrence relation f n ( x ) = x f n−1 ( x ) + f n−2 ( x ), for n ≥ 2, with initial conditions f 0 ( x ) = 0, f 1 ( x ) = 1, Lucas polynomials are defined by the recursion ln ( x ) = xln−1 ( x ) + ln−2 ( x ), for n ≥ 2, with initial values l0 ( x ) = 2, l1 ( x ) = x
In the paper [22] we have introduced the distance Fibonacci polynomials f n (k, x ) given by the following recurrence relation f n (k, x ) = x f n−1 (k, x ) + f n−k (k, x )
We have found a direct formula, a generating function, matrix generators and some identities for generalized Fibonacci polynomials f n (k, x )

Summary

Introduction

Fibonacci polynomials f n ( x ) are given by the recurrence relation f n ( x ) = x f n−1 ( x ) + f n−2 ( x ), for n ≥ 2, with initial conditions f 0 ( x ) = 0, f 1 ( x ) = 1, Lucas polynomials are defined by the recursion ln ( x ) = xln−1 ( x ) + ln−2 ( x ), for n ≥ 2, with initial values l0 ( x ) = 2, l1 ( x ) = x. In this paper, which is a continuation of [22], based on a graph interpretation of the distance Fibonacci polynomials f n (k, x ) we introduce a new generalization of Lucas polynomials in the distance sense. We prove some identities that generalize the classical identities for Lucas polynomials and reveal some Pascal-like relations between coefficients of these polynomials

From the Distance Fibonacci to the Distance Lucas Polynomials

Matrix Generators