


5 Formulae BBP : The techniqueIn 1995, Bailey, Borwein and Plouffe discovered the famous formula
which , as we now know, goes further than it's simple form by giving us a method to find the nth digit of Pi without knowing the previous one. Even better, it revolutionised the view of Pi by putting it amoung the constants that we can generate nearly like a chaotic dynamic system, which Bailey and Crandall turn into their famous conjecture in 2000 [11] linking the concept of normality to the passage of a digit to the next. And with the function , quickly, we will notice that the BBP formulae are a particular case of relations of function class that we call polylogarithms. A nice introduction to those extraordinarly simple but promising function is the work of e Gery Huvent that I adapt here :
5.1 : The polylogarithmes
5.1.1 DefinitionFor , and , we define the polylogarithm of order by
Hence For note
that the convergence is normal of the closed unity disc.
as well
and for example
5.1.2 Remarquable valuesThe first remarquable values are, for where is
Riemann's zéta function
For where . In particular Note 1 We know the exact value of and of
5.1.3 Duplication formulaOn we have , . Which prove the duplication formula, that when the three terms have a direction :
5.1.4 Euler and Landen's formulae for the dilogarithmEuler's Formulae On we have
This equality allow us to prove through differentiation Euler's identity for the dilogarithm
In particular with we get the following result, thanks to Euler
Landen's Formulae We also have the following relation said to be Landen's identity (that we prove using differentiation)
This identity is true on Particular Values With where is the golden number (which satisfies ), since in this case, we obtain
But Euler's formula give us
and the duplication formula
By combining the three results, we get Complex values Landen's indentity can be extend through analysis to With we get
5.1.5 Landen's formula for trilogarithm
This formula can be deduced by Euler's and Landen's identity for the dilogarithm after differentiation.
5.1.6 Particular valuesWe deduce form Landen's identity for the trilogarithm the following remarquable values :
5.1.7 A few classical integralsThe integral representation (35) immediatly give
Similarly
5.2 The links between integrals, BBP formulae and polylogarithms
5.2.1 NotationsLet us start by simplifying notations! It would be great if we could talk about a BBP formula without having to copy down the formula every time... Gery Huvent has
offered to denote the sum , for
Plouffe's formula is now written Which simplifies a lot !
5.3
Integrals and BBP formulae


(62) 
we deduce that if then

(63) 
Similarly,
Now, we have suspicion that it is not this kind of integrals that we would be able to calculate, and so we will not be able to deduce BBP formula, but we should deal with an integral of type . But the whole trick is there !
In fact, if divides where This allows, when we know how to calculate the integral , to deduce a BBP formula. The same kind of remarque is applicable if divides .
Now, lets talk about polylogarithms. The above remarque is true in the case where divides to extract BBP series. But to calculate the integral? Well we can use a decomposition into simple elements of so to go back to known integrals. And that's where polylogarithms come in handy.
We remember that the integral expression of a polylogarithm of order for a non zero complex with modulus less than .

(68) 
This allows, after decomposition into simple elements of , to express an integral of type as sums of polylogarithms.
As well as the obvious relation , the previous paragraph puts light to the decomposition of a rational fraction in the integrals allows to bring BBP formula to polylogarithm. But furthermore, the formula (64), which we have seen the direct link to the BBP formulae if we have a polynomial dividing , can also be written
that is in the form of combination of the functions . In fact, the BBP formulae are nothing other than the combination of functions where the parameter does not move and is the inverse power of an integer.
Hence, the Plouffe's formula
Starting from here, and with order greater than 1 , we have all the bits to link the polylogarithm to the BBP formulae and now the functions .
At this stage, we need to note that the polylogarithms are of the style while the BBP formulae use more series of type . Intuitively, instead of having the parameter as the equivalence let us see, the polylogarithms are able to be "developed" by giving a linar relation between the functions where , basicly the BBP formulae. In this form with the parameters of the function , we can see that this kind of generalisation of the arctan since we don't have anymore but more generaly .
This formulation has showed us that we can in base and calculate any digits of constants resulting from this kind of series, without knowing the previous. Why ? I will explain it a bit later on the page consacred of Plouffe but in an intuitive way, we can notice that the development in base 10 of a number is with an integer between 0 and 9. So by considering the series of the kind we are not far from the development in base 2 ! All we need is to calculate the decimals in a certain place of .
And this is what happened not only with but also with loads of other constants around , the Riemann function etc...
Futhermore, we can notice that they are rational series and so they are perfect target for simple calculation of the decimals of Pi. We know more immediatly that thanks to the term of the series how many decimals we can obtain in practise with n terms of the series. In fact, with decimales, in that case we obtain decimales (and even a bit more if we take into account the term in ).
Alternating intensive phases of exploration with more calm phases, the search in that direction was very fertile! A few elements in the following paragraphs
In a synthetic way, Gery Huvent put forward a few integrals useful to the decomposition that the previous paragraph talk about. We will present it here, with their application to the research into BBP formulae.This text takes with a few adaptations the article by Gery Huvent [12] which can not be more fundamental!