New formulae of BBP-type with different moduli

Abstract

a b s t r a c tInspired by a method used by Bellard in 1997, we establish a trigonometric expansion for the function arctan a x−1 , and then utilize this expansion to derive numerous BBP-type formulae that converge, in most cases, to well-known irrational constants.

The golden ratio, Fibonacci numbers and BBP-type formulas

“…We now present some interesting degree 1 base φ formulas. φ−nary BBP-type formulas for π were also derived in references [4,6,7]. The formulas presented here are considerably simpler and more elegant than those found in the earlier papers.…”

“…Results for arctangent identities involving the Fibonacci numbers and related sequences can also be found in earlier references [1,2,3] and references therein, while results for φ−nary BBP type formulas can be found in references [4,5,6,7].…”

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.…”

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

“…A formula of this type is now called "BBP-type formula". After that, many researchers denoted themselves into this field; Chan [10] gave BBP-type formulas involving the golden ratio; Adegoke [11,12], Bailey-Borweins-Plouffe [13,14], Chan [15,16], Wei [17], Zhang [18,19], Guillera [20], and Takahashi [21][22][23] have found many new BBPtype formulas.…”

“…Some researchers find new formulas by this method; Adegoke [11], Wei [17], and Chan [10,15,16] gave new BBP-type formulas by integral calculus, with some of the formulas involving the golden ratio; Zhang [18] discovered many new BBP-type formulas with different moduli by the expansion of arctan[a/(x − 1)]; Bailey [26] made a systematic summary of the origin and development of BBP-type formulas. It is especially worthwhile to note that the inverse tangent function is closely related to the second method, useful infinite series can be obtained from the study of it [18]. Indeed, the well-known Gregory series can be seen as the simplest series of this type, and it is the infinite series of the inverse tangent function.…”