


6 Formulae BBP in base 2 : ,, in
6.1 The considered integralsIn the worry of decomposing our integrals in simpler to calculate integrals, we are interested to
the following "prime" integrals :
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 methodThe 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 , , andIn the case where , we get, by a calculation of integrals
6.3.1 BBP formulae applications forSo to get some formula for we
impose the following relations : So to get some BBP formula with few terms, we can in a first of all fix , which gives
We then consult the denominator table. To simplify a sum that calls and we can write each integral under the form Hence
where is a polynomial whose coefficients depends on and . If we fix and we get the formula by AdamchikWagon (cf [6] )
We choose the other coefficients so to cancel
out the most coefficients of (by
imposing ). We can also look for a denominator in by fixing the best choice
seems to be In fact it turns out that the last formula can be simplified to , which gives the alternated formula :
and gives
We can explain by
writting that And gives
6.3.2 BBP formulae for andWe can apply the same method to obtain some BBP formula for , and .
6.4 Case of polylogarithms of order : Formulaes of order 2We 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 : andA classical result by Euler is
Which allows us to write that
Kummer's equation for the polylogarithm of order is written (cf [4]) The inverse formula is
and finally the duplication formula in the general case :
By using the inverse formula for and and the duplication formula , we get By duplication, we also have
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
But where is Catalan's constant. Hence
6.4.2 Calculation of andProposition 2 We have
Proof. Kummer's equation with gives the equality which gives straight
away the wanted result because Proposition 3 We have
and
Proof. Kummer's equation for
gives, considering
similarly, Kummer's equation for give
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.
which gives with and
then with and
and allows us to easily conclude. _
6.4.3 Calculating relation between andKummer's equation for and and for and gives two equality which when added gives Which is
and allows us to confirm that
which gives us
i.e.
6.4.4 Application to the determination of BBP formulaeWe 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 : 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. This equality allows to give the general formula with parameters
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 ).
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
which gives the following equality
or
This equality was already mentioned by Plouffe
in [1] 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. 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.
Notice The integrals equality allows to write in
different ways as the sum of BBP formula. 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 )
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 . 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
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 formulaeIn 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 :
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 3The 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
A classical formula allows to state that
and by definition
6.5.1 Calculation ofLanden's equation for the trilogarithm is (cf
[4]
) Applied to we get (with )
result implicitely contained in [3].
6.5.2 Calculation ofAs 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
6.5.3 Relation between and values ofWe 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
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
and gives the relation
which gives
and furnish
6.5.4 Calculation foSimilarly 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 formulaeLet 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 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 . This rquality can be written differently. 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) 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
The simplest BBP formula is obtain for We can also use the following : Finally the simplest formula for a denominator
in is obtained with 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
With , we have the formula with termes This relation is noteworthy, in fact the
equality (204)
is obtain for . Then Finaly the simplest formula with a denominator
in is obtain with Formulae for We have the general formula
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 relationsFor the polylogarithm of order and we only have one tool left, this is Kummer's equation. It is written, with and the inverse formula
6.6.2 Application of the determination of the BBP formulaWe now consider the linear form . With the previous formula giving
6.6.3 Formula for andFor 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 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 relationsBroadhurst 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 apply then the same method as that for polylogarithm of order 4.
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 formulaeWe now consider the linear form . The previous results gives
6.7.3 Formulae for andWe deduce from this some formulae for the
constants and .
We remember that the polynomials are defined by .
with
with with
6.7.4 Simplification of those formulaeThose formula can be simplified by making the term appear. For example which gives Similarly
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