


13 Harmonic SeriesThe BBP formulae are expressed for example as linear combination of those hypergeometric functions but even more things await us! Because thanks to integral representation, we can also in fact obtain harmonic series of the same form as all those that we just found. The harmonic series are those which the terms contain the harmonic sum .
13.1 Links between harmonic series and the polylogarithmsWe have seen that the BBP or factorial formulae are integral combinations of the kind . Let us now consider the integral equivalent to the serie's term of the harmonic sum . According to the product formula by Cauchy for which two absolutly convergant series and
where , by choosing and we get and so
You can start to see what I mean... :) As soon as an integral uses this kind of formula, we will have some harmonic lurking in a corner. Here, we find one immediatly in the formula
With we had some polylogarithm (voir 35), but this doesn't matter, we divide the expression 588 by x ! And then integrating between and , we have But, since , we finaly obtain that
Impressive, no? obviously, this remind us of many things, we are swimming in a sea of polylogarithm and logarithm... We can study in more generality the harmonic series of this kind and the formulae that follows from them by introducing the notations by Gery Huvent (him again !). All the following formulae which are not yet known can be credited to him !
13.2 Study of and of
13.2.1 Definition, remarquable relationsWe let Those series have a convergence radius of 1. The convergence take place on the border as soon as (because ). Then Proof. hence Similarly
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13.2.2 Calculations of certain functionsA usual expansion (donne by Cauchy's product) gives
Then by integrations
and
Before carrying on, let us examine this equality. For the serie defining converges, but the right hand side of the previous equality does not. However with the formulae by Euler and Landen, we get The last equality gives us an expression of
valid for . By analytical expansion, we know
that the different expression obtained coincide where they are defined simultaneously. We deduce from the calculation of that
then by integration that Finaly
and hence We deduce from it that
Then by a differentiation calculation, we have which gives for real and and finally
13.3 Applications to the calculation of certain seriesWe use the previous results with some rightly chosen values for
13.3.1 WithThe values are more easily obtained with the Beta functions, for example
13.3.2 WithWe obtain by taking the value of the functions and in
By combining, we get
Calculation of for We can then get
Calculation of for To sum up : We can combine those results and obtain
13.3.3 WithThe convergence for the serie is no problems for we then get
For we find that Which gives us
Those equalities are justified because
alternating series converges and with of theorem like Tauber, we
conclude. which gives by subtracting them and with We also have
13.3.4 WithWe then get by considering the real and imaginary parts The convergence is assured by making for real and by passing through the limit in with Abel's lemma. With We get taking into account the value of and Landen's equality in But, we have proved in [12] (calculation of )
which gives us
and allows us to find with
With The calculations are a bit more complicated, we use this time the equality (658), then the inverse formula for the polylogarithm of order which gives
as well as
which correspond to the calculation of in my paper ”formules BBP”, we get We apply the same substitution for and . We then obtain by considering the real part (the imaginary part does not give anything useful) By combining the different equations obtained, we can deduce
13.3.5 WithWe use in this case the following result. (mgl means "multiple by Glaishers” and mcl means ”multiple by Clausen”) then where is the nth polynomial by Bernoulli. Then the duplication formula
with which allows to express and with the help of and . We then obtain the following results : With
the convergence of this serie is justified by
summation in different parts.
but
where or
With
which gives us
the convergence of this serie is justified by the summation by parts.
which gives
we also have
With we obtain the following formula
the other make some intervine which gives with
13.3.6 WithBy using the equality
and it's conjugate, we have whose convegence is assured by packets. By combining with (675), we obtain
13.3.7 WithFor We get with the two following formulae which are remarquables and with the two equality
With the functions we have With The function gives immediatly
and gives
13.4 Generalisation
13.4.1 Euler's sumsIf we define (partial sum of
), we have Euler's theorem. We have some relations with the polylog : . By integrating we have for example
13.4.2 A formula combining Harmonic and combinationHere is a little serie that I've found recently in novembre 2001, mixing the combinations and the sums harmonic ! We can maybe find a more simple proof, but I do find this one quite elegant in the end. Ok, it's a particular case, I don't know if we can find other series of this kind (and numerically I have yet to find one), but it is maybe worth the effort of searching! Proof. Let . Then hence
hence in particulae in and by regrouping the two series containing some , we obtain
which simplify a bit more the work ! The trick unfortunatly does not seem useful because the serie in is difficult to calculate it seems to me.... Let us find another way. Note that we have i.e. . Hence using integration by parts. Hence since using integration by parts, just simply, on the convergence radius of (). We now need to properly integrate this serie to find a term in :
let us add the term for :
We now use 705 to obtain
hence finaly
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13.4.3 An other formulaWe also find some formulae with some special harmonic sums such as Bradley's formula :
or even this representation of which present some troubling similarity with the previous one !
The proof is available in [10].
13.4.4 Harmonics of harmonics !According to the Gradshteyn [9] (1.516), we can immediatly obtain
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