Linear Properties of Generalized $n$-step Fibonacci Numbers

TL;DR

This paper explores various mathematical properties of generalized n-step Fibonacci and Lucas numbers, including recurrence relations, summation identities, and generating functions, with explicit examples for small n.

Contribution

It introduces new properties and identities for generalized n-step Fibonacci and Lucas numbers, expanding understanding beyond classical cases.

Findings

Derived new recurrence relations for n-step Fibonacci numbers.

Established binomial and double binomial summation identities.

Provided explicit examples for small n values.

Abstract

We present numerous interesting, mostly new, results involving the $n$-step Fibonacci numbers and $n$-step Lucas numbers and a generalization. Properties considered include recurrence relations, summation identities, including binomial and double binomial summation identities, partial sums and ordinary generating functions. Explicit examples are given for small $n$ values.