A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers

TL;DR

This paper introduces a symmetric recurrence-based algorithm applicable to hyperharmonic, Fibonacci, and Lucas numbers, deriving explicit formulas, generating functions, and defining new hyperfibonacci and hyperlucas numbers to explore their interrelations.

Contribution

It presents a novel symmetric recurrence algorithm and new hyperfibonacci and hyperlucas numbers, expanding the analytical tools for these sequences.

Findings

Derived explicit formulas for hyperharmonic numbers.

Established general generating functions for Fibonacci and Lucas numbers.

Explored relations between ordinary and incomplete Fibonacci and Lucas numbers.

Abstract

In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated.