On the Sequences of $(q,k)$-Generalized Fibonacci Numbers

TL;DR

This paper introduces and analyzes a new family of $(q,k)$-generalized Fibonacci sequences, extending existing Fibonacci and Pell sequences, with formulas, asymptotic behavior, and binary sequence characterizations.

Contribution

The paper defines the $(q,k)$-generalized Fibonacci sequences and derives formulas, asymptotic properties, and binary sequence characterizations, expanding the understanding of generalized Fibonacci-type sequences.

Findings

Derived a Binet-style formula for the sequences

Analyzed the asymptotic behavior of the dominant root

Characterized initial terms using binary sequences

Abstract

In this paper, we consider the new family of recurrence sequences of $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell numbers. We shall obtain a Binet-style formula and study the asymptotic behavior of dominant root of characteristic equation. Moreover, we shall prove some auxiliary results about these sequences. In particular, we characterize the first $(q,k)$-generalized Fibonacci numbers in terms of binary sequences.