


7 fixed integers,, : BBP in base 3So 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 forWe 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 :
7.2 Formulae of order : integrales withThere exist reals such that ( the two members are linear form in
decompose on two different base of the dual of )
which under integral form, give
and correspond to the equality . We can deduce the value of . But we have where
By using We deduce from it
and
From (273) and (277) we get the values of and and the equality
Note 7 is another relation proved by Ramanujan .
7.3 Formulae of order 3 : Integrales with lnHere 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
Proof. Kummer's equation for the polylogarithm of order is written With and we obtain Landen's equation is
Applied to and to , it allows to express and in functions of and . The relation
applied to and allows to simplify the left member of (289).
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
In particular, we have with
which gives the equality
Note 12 We can also take and to a change of variables which leads to
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
Proof. Landen's equation applied to where is a root of (the other root is such that ) give
By using the equations (293)
and (299), we deduce
from it that
And hence
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