On the representation of k sequences of generalized order-k numbers

TL;DR

This paper introduces k-generalized order-k numbers, explores their relationships, and provides determinantal, permanental, and Binet's formula representations, advancing the mathematical understanding of these sequences.

Contribution

It presents new relations and matrix-based representations for k-generalized order-k numbers, including a Binet's formula for generalized order-k Pell numbers.

Findings

Derived relations between i-th and k-th sequences.

Provided determinantal and permanental representations.

Established Binet's formula for generalized order-k Pell numbers.

Abstract

In this paper, we define k-generalized order-k numbers and we obtain a relation between i-th sequences and k-th sequences of k-generalized order-k numbers. We give some determinantal and permanental representations of k-generalized order-k numbers by using various matrices. Using the relation between i-th sequences and k-th sequences of k-generalized order-k numbers we give some determinantal and permanental representations of i-th sequences of generalized order-k numbers. In addition, we obtain Binet's formula for generalized order-k Pell numbers by using our representations.