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### 6 Formulae BBP in base 2: ,, in

#### 6.1 The considered integrals

In the worry of decomposing our integrals in simpler to calculate integrals, we are interested to the following "prime" integrals :

 (83)

 (84)

In [3], Broadhurst establish some relation between the different sums of polylogarithm. However he does not seem to consider the link with integrals. We will try to do so.  Which is why we introduce the following integrals :

The link between the and the is the following :

#### 6.2 The method

The calculation of the integrals will give linear combination of constants of order like or thanks to their expression under polylogarithm form of order . But furthermore, we can obtain BBP formula with the  by using what Gery Huvent calls the denomination tables and which are just the expressions in the form of integrals whom we have seen the direct expression under BBP serie form with the formula (69). We just need to obtain BBP series for the precise constant that we are interested in, which boils down to use a certain linear combination of . For this, we intoduce normally the linear form . Then we impose some relation between the so to cancel out the coefficients in fronts of the unwanted constants. We then get a BBP serie for the remainder, or or a lot more other things! Here is the denominator table. The method is then detailled with an example.

where is a polynomial with integers coefficient which depends both of and of the chosen denominator.

#### 6.3 Formulae for , ,  and

In the case where , we get, by a calculation of integrals

##### 6.3.1 BBP formulae applications for

So to get some formula for we impose the following relations :

.
So we get

So to get some BBP formula with few terms, we can in a first of all fix , which gives

 (113)

We then consult the denominator table. To simplify a sum that calls and we can write each integral under the form Hence

 (114)

where is a polynomial whose coefficients depends on and . If we fix and we get the formula by Adamchik-Wagon (cf [6] )

 (Adamchik-Wagon)

The choice of give the formula by Plouffe (61). We can now look for a denominator in . We take then the formula (111) and we simply fix .

We choose the other coefficients so to cancel out the most coefficients of (by imposing ).
The best choice seems to be
which gives

We can also look for a denominator in by fixing the best choice seems to be
which gives

In fact it turns out that the last formula can be simplified to , which gives the alternated formula :

 (Bellard) (119)
Finnaly if we considered the formula (111) in all generality, it can be written , we just need to choose the right . The choice that seems to lead to a formula with the least terms possible is
and gives
 (120)

We can explain by writting that
.
To finish off, the choice of leads to

And gives

##### 6.3.2 BBP formulae for and

We can apply the same method to obtain some BBP formula for , and .

#### 6.4 Case of polylogarithms of order : Formulaes of order 2

We can notice that for order 1, that is for the BBP formulae giving or for example, we had to deal with integrals with rational fractions. If we now want to get BBP series giving or or , we need to introduce a logarithm to the numerator of the integral. More precisely, for a BBP formula of order , we need to consider integrals of type . This is due to the fact that we then obtain polylogarithm combination of order .

This is what was done once more by Gery Huvent [12], noting as well that the first series were found by Plouffe and that Broadhurst [3] provided a few as well. The interest in order 2 is to be able to find BBP series for a famous constant which is Catalan's constant defined by . This shows, if you are not convinced, that Catalan's constant is "homogenuous" to an order 2, that is that it is without doubt of the same nature as or concerning the spread of its digit in base 2 or 16.

##### 6.4.1 The classical expression: and

A classical result by Euler is

 (123)

Which allows us to write that

 (124)

Kummer's equation for the polylogarithm of order is written (cf [4])

The inverse formula is

 (Formule d’inversion)

and finally the duplication formula in the general case :

 (Formule de duplication)

By applying Kummer's equation for then , we get two equality which added gives

By using the inverse formula for and and the duplication formula , we get

By duplication, we also have

 (128)

We therefore deduce

Similarly, Kummer's equation for and , gives two equality which when taken away fives a new equality. By then using the inverse formula for and and the duplication formula for and we obtain

 (131)

But where is Catalan's constant. Hence

 (132)

##### 6.4.2 Calculation of and

Proposition 2 We have

 (133)

Proof. Kummer's equation with gives the equality

which gives straight away the wanted result because
,

by the duplication formula and Kummer formula for gives  _

Proposition 3 We have

 (134)

and

 (135)

Proof. Kummer's equation for gives, considering

 (136)

similarly, Kummer's equation for give

 (137)

With the help of the inversion formula

Which allows us to conclude that

and

So we just now need to calculate the last sum of polylogarithm. But we have gain in simplicity because those logarithm uses roots of unity.
We therefore use the multiplication formula

 (Formule de multiplication)

which gives with and

 (144)

then with and

 (145)

and allows us to easily conclude.  _

##### 6.4.3 Calculating relation between and

Kummer's equation for and and for and gives two equality which when added gives

Which is

 (147)

and allows us to confirm that

 (148)

If, instead of adding them, we substract them, we get

which gives us

 (150)

i.e.

 (151)

##### 6.4.4 Application to the determination of BBP formulae

We now consider the linear form . Taking into account the equalities (123), (124), (129), (130), (132), (133), (134), (135) and (148), we have

Formulae for So to obtain BBP formulae for we fix the following equality :

so to obtain the equality

We now just need to use particular variables so to obtain simple formulae.

A few simple alread known formulae for

We obtain those formulae by choosing the integrals who give an denominator of a degree less than in the correspndance table.
Let us one last time show details an example :
So to obtain a denominator of the form of smaller degree (in that case ), we choose such that we cancel down and . OSo we then get

This equality allows to give the general formula with parameters

 (160)

So not to put too much on this page only the formulae with parameters that correspond to the denominator in will be given. There exist some who are assoiciated to other denominators (for example ).
Let us look again at the equality (159), if we choose to fix and this equality becomes

 (161)

We can look at this equuality under the form of the sum of BBP formula.

We can bring vack this integral to a denominator of the form with the help of the corresponding table so to obtain

 (163)

which gives the following equality

 (164)

or

 (165)

This equality was already mentioned by Plouffe in [1]
The choice of gives

which gives the following formula thanks to Plouffe :

A few simple and new formulae

An other solution consist of keeping the integrals which gives denominator of the form of high degree but adjust the parameters so to gave many nul coefficients in the BBP formulae.
A few good choice seems to be the following  :
which gives the formula to terms

The polynomial having only odd powers, this formula can be simplified to give

which gives the formula to terms

which gives the formula to terms (notice the ) :

If we are interested in formula with the least term, let us state that other formulae with terms ( ) and with terms () exists.
We can even look for alternating series, for example
gives

 (Huvent)

Notice

The integrals equality allows to write in different ways as the sum of BBP formula.
For example, the result by (Huvent) ( ) gives the integral equality

Which can be written

Formulae for the constants and For

The same method leads to the general formula for (when we impose a denominator in )

 (Huvent)

By adjusting the coefficients and we have formulae with termes of the form

One of the simplest seems to be

An other formula with termes is obtained for

To finish gives

For Catalan's constant

Similarly, by imposing a denominator in we obtain

The most interesting case is obtain when all the are zero except which we let be equal to . We then obtain

The interest in this formula resides in the coefficients of which are all powers of .
The choice of allows to write
where has non-zero coefficients.
To finish gives where has non-nul coefficients.

Warning 4 It seems that I was the first to have experimentaly discovery a real formula  for (Mai 2000) without being able to find proof. The discussion between me and  David Broadhurst with no doubt, but that's not very important...

For

We obtain

 (173)

The simplest case is given by which leads to

The choice of leads to where has non zero coefficients.

##### 6.4.5 A few composite formulae

In the determination of BBP formulae, we have systematicly cancel down the coefficients of and . If we then decide to keep those term, we can obtain among those possible formulae, the following result :
give

 (174)

which can be simplify to

This formula is equivalent to the formula (167). If we apply this idea to Catalan's constant, we obtain with the equality

Who under serie form gives the two following equality :

The same idea leads, with to

and with to

This kind of formula has not been systematically researched.

#### 6.5 Cases of polylogarithms of order 3

The order 3 introduce of course formulae giving and other but mostly the famous constant proved irrational by Apéry in 1978 [14] by developping a continued fraction of a factorial formula which will be mentioned later on.

This constant stays never the less mysterious, and the BBP formula intuitively shows that this constan is very likely not to be very different from a point of vue of the orderning of it's decimals and hence it's complexity.

Kummer's equation for the triologarithm is

and the inverse formula

 (181)

A classical formula allows to state that

 (182)

and by definition

 (183)

##### 6.5.1 Calculation of

Landen's equation for the trilogarithm is (cf [4]

Applied to we get  (with )

 (185)

result implicitely contained in [3].
We deduce by the duplication formula that

 (186)

##### 6.5.2 Calculation of

As for the calculation of we use Kummer's equation with then with We add then those two equation we obtain. We simplify those equations with the help of the valuer of and the equality . This allows us to state that

 (187)

##### 6.5.3 Relation between and values of

We take the first two equation for the calculation of that we substract this time. We then use the inverse formula with and so that we make appear the term . Finaly one last application of the inverse formula with and with leads to

Which prove that

 (189)

If we use Kummer's equation with and then with and we obtain two equality that we substract. We simplify the obtained result with the inversion formula applied to and so to obtain

This equality is equivalent with the help of integrals to

 (191)

and gives the relation

 (192)

If, instead of substracting the equality obtained previously, we add them, we get :

which gives

 (194)

and furnish

 (195)

##### 6.5.4 Calculation fo

Similarly to the calculation of Kummer's equation for the polylogarithm of order with , then the inverse formula with leads immediatly to

##### 6.5.5 Application to the determination of BBP formulae

Let us now consider the linear form . Then the previous result allows us to state that

Formulae for If we look for formulae for we cancel down the coefficients of the constants
... to obtain

We can not choose which mean we have to use a denominator in for the BBP formulae giving . In all generality, we hence obtain a BBP formula for with two parameters ( and ), formula that the reader can establish. The simplest formula is hence obtain for and with termes. So to wrie it, we introduce the polynomials defined by , for example .
We then have

This rquality can be written differently.
In fact
But and
where has non-zero coefficients in general, and only if or or .
If we apply this here, we have
which gives

This formula can also be written as

Formula for To obtain a simple BBP formula for , we apply the same idea as that for (the problem is the same, we need to use which gives a denominator in and gives an expression with two parameters)
We then obtain with

This formula can also be written

Compare with those obtain for .

Formulae for For we obtain, if we look for a denominator in

Which give the general formula

 (Huvent)

The simplest BBP formula is obtain for

We can also use the following :

But and

If we then impose in (203), we have
which gives

Finally the simplest formula for a denominator in is obtained with
and contains termes.

Note 5 Similarly here, I thought I was the first to give a formula for whose proof, a lot less simple but on the same model, was finished in june 200 with the help of Raymond Manzoni [5].

Formulae for We obtain the following general formula

 (Huvent)

With , we have the formula with termes

This relation is noteworthy, in fact the equality (204) is obtain for
This leads to fix

Then

Finaly the simplest formula with a denominator in is obtain with
and has termes.

Formulae for We have the general formula

 (Huvent)

and the simplest formula is ontained for

And the one with a denominator in is given by

#### 6.6 Cases of polylogarithm of order 4

##### 6.6.1 The relations

For the polylogarithm of order and we only have one tool left, this is Kummer's equation. It is written, with

and the inverse formula

 (214)

We then so apply the following method :
- Using Kummer's formula with a particualr couple
- Eliminating the polylogarith with argument modulus greater than with the inverse formula
- Eventual simplification with the help of the duplication formula or some value of in and .

• With we get Which with the integrals and and becomes
• With we get

We obtain a second equality with conjugating (or with ). We add, or substract the two equality obtained and by replacing by or , this gives

and

• With we obtain  (221)
We then use the multiplication formula  (222)

which with allow us to confirm that

 (223)

and that

 (224)

• Finaly, with , we obtain  (225)
By considering the real part and the imaginary part of this expression, we have respectively

##### 6.6.2 Application of the determination of the BBP formula

We now consider the linear form . With the previous formula giving

##### 6.6.3 Formula for and

For The simplest formula (and the only associated with a denominator in ) is obtained for and gives

There exist a formula with two parameters with a denominator in , the simplest is given by

For The simplest formula is obtained with
and gives

For Wiht we get

The case of and It is not possible to determine formulae for those constant, we can only find two independant relation which are :

This shows that we only need to find a formula for one of those three constant and then we can deduce one for the two others.

#### 6.7 Case of polylogarithm of order 5

##### 6.7.1 The relations

Broadhurst in [3] shows relations between the integrals and with the help of Kummer's equation (relations to . Then discover two equality through numerical mean (relations and ). He then prove the relation by using the hypergeometric series and Euler's sums. He deduce from it four equalities for (relation )
We can establish those four equalities directly with the help of Kummer's formula which is written with

We apply then the same method as that for polylogarithm of order 4.

• With we obtain, taking into account

wich gives us

Which can be simplified to, by using

• With we get

which is equal to

• With , we get

We simplify this equality with the help of and the multiplication formula , which with allows us to have .
Hence

• With we obtain

which gives us

Note 6 The relation proved by Broadhurst ([3]) correspond to the calculation of The calculation given here seems a lot more simple.

##### 6.7.2 Application of the determination of the BBP formulae

We now consider the linear form . The previous results gives

 (266)

##### 6.7.3 Formulae for and

We deduce from this some formulae for the constants and . We remember that the polynomials are defined by
We hence obtain
with

with

with

with

##### 6.7.4 Simplification of those formulae

Those formula can be simplified by making the term appear. For example

which gives

Similarly

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