Some Properties of \(k\)-Order Gaussian Fibonacci and Lucas Numbers

Abstract

In this paper, we define and study the \(k\)-order Gaussian Fibonacci and Lucas numbers with boundary conditions. We identify and prove the generating functions, the Binet formulas, the summation formulas, matrix representation of \(k\)-order Gaussian Fibonacci numbers, and some significant relationships between \(k\)-order Gaussian Fibonacci and \(k\)-order Lucas numbers, connecting them with usual \(k\)-order Fibonacci numbers.