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### 7 fixed integers,, : BBP in base 3

So many results already ! But it's not over yet... in fact, even if we can not BBP formula for all bases, an important class of BBP formula was obtained in base 3, which correspond to taking . Here again, Gery Huvent explained the best the relations in [12].

#### 7.1 Formulae for

We take the same kind of integrals than that of base 2, but this time it's the 3 that will be omnipresent in the polynomials.

An elementary calculation shows that

By cancelling the coefficients of and we obtain a formula for  :

The following particular case is the most elegant  :

 (Huvent)

#### 7.2 Formulae of order: integrales with

There exist reals such that

( the two members are linear form in decompose on two different base of the dual of )
Plouffe showed that

which under integral form, give

 (271)

and correspond to the equality . We can deduce the value of .

But we have

where
We deduce from it

 (273)

But Ramanujan proved that

 (274)

By using

We deduce from it

 (276)

and

 (277)

From  (273) and (277) we get the values of and and the equality

 (278)

Note 7 is another relation proved by Ramanujan .

Note 8

We deduce

Note 9 , with and and changing variables , we get

 (281)

and the equality

 (282)

Note 10

 (Huvent)

#### 7.3 Formulae of order 3: Integrales with ln

Here again, the order 3 introduce some formulae giving and other but also .

There exist reals such that

The calculation of is elementary because

Proposition 11 We have

 (285)

This relation is equivalent to

Proof. Kummer's equation for the polylogarithm of order is written

With and we obtain

Landen's equation is

 (290)

Applied to and to , it allows to express and in functions of and . The relation

 (291)

applied to and allows to simplify the left member of (289).
Then all there is left to do is use the two following equalities

to conclude.  _

With and (285) can be written

If we replace by it's value compared to the right hand side of becomes a polynomial of degree in whose coefficient of is equal to . Hence does not depend on . After calculation, we get

 (294)

In particular, we have with

 (295)

which gives the equality

 (Huvent)

Note 12 We can also take and to a change of variables which leads to

 (296)

and to the same equality as for the series.

We just need to calculate the exact value of and .

Proposition 13 We have the equality

 (297)

Proof. Landen's equation applied to where is a root of (the other root is such that ) give

 (298)

But and . We deduce that

_

By using the equations (293) and (299), we deduce from it that

And hence

 (302)

Note 14 With we get

 (303)

and the following BBP formula