11 Centrals binomial series
An interesting particular case and yet not to har on factorial series involve the introduction to the central binomials coefficients in the series. After a few notes on those series in the first section we will plunge into serious stuff with several formulae and their proofs!
11.1 Combinations inversion
A funny thing is that we can obtain some series giving the same result but in one case with the central binomial coefficient on the numerator, while in the other on the denominator! This also confirrm why we obtain the same kind of results with a serie with a combination and one without:
Let us take the example of the function: first of all, we have the classical formula
We know from Euler (see Euler's page) that the arctan function can also be expressed as
Finaly, the coupled relation with give
A relation between the different series and the central binomial coefficients is hence the arctan :
Suprising, no ? ! !
By deriving the serie from the center along , we can for example find the expression under the combination form of .
And so starting from this serie, we can find the combination expression of serie of type Machin under the form of combination of binomial formulae! We will see in the following section the 11.3factorials.
We can anyway signal for the arctan function
The proof is obvious by decomposing the serie in even and odds...
Furthermore, we also have, still playing of the developpement of power series
Finally, we have according to the expression 13 and 11
which is ammusing as well !
11.2 Useful development
We remember the following development of power series which will be useful for the calculation of certain series and integrals. :
which, by composing by or and by integrating, give respectively
furthermore, Euler's developpement given under it's form
that we can hence proof by the method of differential equations.
11.3 First direct formulae
The four last developpement of the previous section give then classical series, with
we can obtain, for example
With the developpement of , we obtain
and obtain pther series such as
By differentiating again , we have
On the same model, Gery Huvent offered what seems an open conjecture, having not read stuff on this subject...
11.4 Formulae of greater order
The study of series of the kind or , for complicated things a lit, because those sums comes from succevise integrations of functions where . In fact, we will notice that
and so on to go back for example up to .
but of course, those functions don't let themself be easily integrated! And we often need to have a lot of patient and a few trick up our sleeves to see the end of it.
However, the results are there. We know Euler's formula
who I give a proof to the paragraph dedicated to 10.2calculus and the method of convergence acceleration by the finite difference, invented by Euler himself!
The greater order, it's a formula such as
and the formula by Comtet dating of 1974 whose proof was tedious...
Obviously, this intrigued mathematicians to know wether a similar formula existed for order 5! Unfortunatly, we now know that this serie does not give the wanted result since if we recall , we have according to 
But this doesn't stop the higher order from being interesting.
11.4.1 A first formula with proof
I noticed in august 2001 that
which is transformed into hypergeometric form into
It must be one of the simplest factorial formula and the presence of the make it seems like a BBP formula. The proof is complicated and linked to the folowing integral calculation
because the development into power serie of give
which we know how to calculate the first two part.
We hence let more generaly the following integrals :
Note that we can clearly brings this integral back to 9.4.1in the sense where and hence
for . There no coincidence !
The result that interest us is
which is equivalent to the serie through an immediate change of variable in 414.
Raymon Manzoni, then Gery Huvent , contributed to give a proof for this formula, which I will present to you now, and also including a proof of the formula by Comtet done by Gery Huvent .
Proof. Consider, for integers and the function
this function is holomorph on
If disign a primitive of on the domain containing the path we have
The problem hence goes back to the calculation
of a primitive !
We therefore obtain a second expression of by comparing with it what we already obtained, for , we find again simply the value of . This is nothing new, because we know the developpement in Fourier siries of the nth polynomial of bernoullli. This developpement gives the equality
The primitive of which is nul in is
We can refind that and that
which allows us to find Euler's formula for the dilogarithm
Note 22 We can simplify this equality with Landen's formulae and find the classical result
result that we obtain with the developpement into Fourier's series of and Parseval's formula.
This equality then gives with and a DSE
This equality allows several applications :
obtained by integration by parts and changing
variables because .
that we developpe like in the previous example, we find
With and taking into account the value of
By developping the square under the integral and by taking into account the previous calculated values of it turns out (Oh miracle!) that we get
With the result is less simple, however we have
By combining with 433, we have
Now the trick is to use this approach to developpe a particular method and as often we need to go through equivalent integrals, who we just saw a first example. All of this on an idea by Gery Huvent :
11.4.2 Calculation of a primitive of
For , we introduce the function
Proof. We differentiate the right hand side,
Then in the right hand side is equal to . _
Application :Intégrales of kind Dirichlet Consider the equality (446) with we obtain the expression of the integral with the help of polygarithms. An integration by part (where we integrate the cotangent) gives
We then obtain the following equalities
With we get
For the folks mathematics We then easily show that
The minimum of the trinomial is obtain in , funny !
11.4.3 Uses of
Uses of the primitive for We have for and
Hence for or we obtain
The change of variable then gives
As an application, we obtain for
As another application, gives
Uses of the primitive of We have for and
the change of variabl gives
As an application, we have with
11.4.4 Uses of
Uses for the primitive of We have for and
by change of variables, we obtain
With , we get
It's with this last equality that Apery proved
the irrationality of .
Uses of the primitive of By the same kind of method we obtain
which with gives
By combining those last two equalities with the one found by Apéry (474), we get
11.4.5 Other formulae
Starting from the result stated in the Gradshteyn  (4.241), we get the formula
Nice formula, putting in evidence the arrival of a constant linked to elliptical integral of first sort, not bad...
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