On (k,p)-Fibonacci Numbers

Natalia Bednarz

doi:10.3390/math9070727

Abstract

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too.

Highlights

IntroductionFor special values of r and bi , i = 1, 2, · · · r, the Equality (1) defines well-known numbers of the Fibonacci type and their generalizations
By numbers of the Fibonacci type we mean numbers defined recursively by the r-th order linear recurrence relation of the form an = b1 an−1 + b2 an−2 + · · · + br an−r, for n > r, Citation: Bednarz, N
We prove some identities for Fk,p (n), which generalize well-known relations for the Fibonacci numbers, Pell numbers, Narayana numbers, k-Fibonacci numbers, Fibonacci s-numbers and generalized Fibonacci numbers, simultaneously

Summary

Introduction

For special values of r and bi , i = 1, 2, · · · r, the Equality (1) defines well-known numbers of the Fibonacci type and their generalizations. Numbers of the Fibonacci type defined recursively by the second-order linear recurrence relation were introduced by A. Fibonacci s-numbers (Stakhov [6]): Fs (n) = Fs (n − 1) + Fs (n − s − 1) for integer s > 1 and n > s + 1, with. Distance Fibonacci numbers (Bednarz, Włoch, Wołowiec-Musiał [8]): Fd(k, n) = Fd(k, n − k + 1) + Fd(n − k) for integers k > 2, n > k, with Fd(k, n) = 1 for 0 6 n 6 k − 1. For other generalizations of numbers of the Fibonacci type see, for example, in [11]. We consider a special kind of this equation with the assumption that bi can be rational

Generalization and Identities

Conclusions