New BBP-Type Formulae for π Derived from New Forms of Taylor Expansions of Inverse Tangent Function

Abstract

In this work, we give two new Taylor expansions of arctan(x + ω), where ω represents a finite increment of x. We discover several remarkable infinite series from these expansions by special substitutions. Some of these infinite series give BBP-type formulae.

Logarithmic integrals with applications to BBP and Euler-type sums

“…where b, d and m are positive integers with b ≥ 2, and (a 1 , a 2 , ..., a m ) ∈ Z m . After the discovery of the BBP-type formula for π mentioned above, many authors devoted themselves to find new BBP-type formulas for the other mathematical constants and they provided interesting formulas; see [2][3][4][26][27][28][29]. In the literature there exist different variants of BBP-type formulas.…”

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

“…Other interesting formulas were also recently published, derived from new forms of Taylor expansions of inverse tangent function [19].…”