On the Product Representation of Number Sequences, with Application to the Fibonacci Family

TL;DR

This paper explores how number sequences can be explicitly represented through products, focusing on Fibonacci-related sequences and revealing recursive identities and properties shared among these families.

Contribution

It introduces a general framework for product representations of number sequences and derives new recursive identities for Fibonacci and generalized Fibonacci sequences.

Findings

Sequences can be explicitly represented as products.

Fibonacci and generalized Fibonacci sequences share recursive identities.

New recursions linking these sequences are established.

Abstract

We investigate general properties of number sequences which allow explicit representation in terms of products. We find that such sequences form whole families of number sequences sharing similar recursive identities. Restricting to the cosine of fractional angles, we then study the special case of the family of $k$-generalized Fibonacci numbers, and present general recursions and identities which link these sequences.