On the genesis of BBP formulas

Daniel Barsky (1) , Vicente Muñoz (2) , Ricardo Pérez-Marco (3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Universidad de Málaga [Málaga] = University of Málaga [Málaga]
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for π. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing 0 come from. We also explain what is the observed relation between some BBP formulas for log 2 and π, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transal-gebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants.

Domaines: Variables complexes [math.CV]; Théorie des nombres [math.NT]

Daniel Barsky, Vicente Muñoz, Ricardo Pérez-Marco. On the genesis of BBP formulas. Acta Arithmetica, 2021. ⟨hal-02164612v1⟩