Relations between $π$ and the golden ratio $ϕ$ in the form of Bailey-Borwein-Plouffe-type formulas

Abstract

We provide a family of expressions of π in terms of the golden ratio φ in the same spirit of the formula obtained by Bailey, Borwein and Plouffe for π. Connection with cyclotomic polynomials is outlined.

New series representations for any positive power of $π$ from a relation involving trigonometric functions

“…There are very few known expressions for π m with m = 2k + 1 and k ∈ N * [1][2][3]. We recently published series representations for π 3 [4] and π 5 [5], consisting of a prefactor depending only on the golden ratio multiplied by an infinite sum.…”