The golden ratio, Fibonacci numbers and BBP-type formulas

TL;DR

This paper derives new arctangent identities involving the golden ratio and Fibonacci numbers, introduces novel BBP-type formulas for mathematical constants in golden-ratio-base, and simplifies existing formulas significantly.

Contribution

It presents the first derivation of binary BBP-type formulas for certain arctangents of the golden ratio and introduces simpler golden-ratio-base BBP formulas for key constants.

Findings

New arctangent identities involving golden ratio and Fibonacci numbers

First binary BBP-type formulas for arctangents of odd powers of the golden ratio

Simpler golden-ratio-base BBP formulas for π, log 2, log φ, and √2 arctan √2

Abstract

We derive interesting arctangent identities involving the golden ratio, Fibonacci numbers and Lucas numbers. Binary BBP-type formulas for the arctangents of certain odd powers of the golden ratio are also derived, for the first time in the literature. Finally we derive golden-ratio-base BBP-type formulas for some mathematical constants, including $\pi$, $\log 2$, $\log\phi$ and $\sqrt 2\,\arctan\sqrt 2$. The $\phi-$nary BBP-type formulas derived here are considerably simpler than similar results contained in earlier literature.