On some properties of $k-$generalized Fibonacci numbers

Halim Özdemir; Sinan Karakaya

Vol. 29 No. 2 (2024)

Regular Articles

On some properties of $k-$generalized Fibonacci numbers

Published 2024-10-04

Halim Özdemir
Sinan Karakaya

Halim Özdemir
Sakarya University

Sinan Karakaya
Sakarya University

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Abstract

Some identities between the positive integer powers of the matrices $X$ satisfying the equation $X^{k+1}-(r+1)X^k+rI+\sum\limits_{i=1}^{k-1}(r-1)X^i=0$ and the $k$-generalized Fibonacci numbers are first established, where $k$ is an integer with $k\geq 2$. Then, two main formulas, one of which is related to the $k$-generalized Fibonacci sequence, are given.

Keywords

$k$-generalized Fibonacci numbers, recurrences, polynomials, matrices

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Author Biography

Halim Özdemir

Department of mathematics

Sinan Karakaya

Department of mathematics