Abstract

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

Highlights

  • Introduction and Preliminary ResultsLet k ≥ 1, n ≥ 0 be integers

  • Sequences defined by the (k + 1)-st order linear recurrence relation

  • In this paper we introduce another type of distance Fibonacci polynomial using the power of the variable x, which relates to the order of the recurrence equation

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Summary

Introduction and Preliminary Results

Some results concerning applications of graphs in studying known sequences of the Fibonacci type were given, for example, in [11,12,13,14,15,16,17] Motivated by their results in this paper we use graph methods to give properties of generalized Fibonacci polynomials. Staton defined Fibonacci polynomials of graphs being the total number of independent sets in G [Kx ]. In this paper we introduce another type of distance Fibonacci polynomial using the power of the variable x, which relates to the order of the recurrence equation This implies that we can only use known methods partially.

Connections with Pascal’s Triangle
Matrix Generators
Conclusions
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