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### 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

 (372)

We know from Euler (see Euler's page) that the arctan function can also be expressed as

 (373)

Finaly, the coupled relation  with   give

 (374)

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

 (378)

Finally, we have according to the expression 13 and 11

 (379)

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. :

 (380)

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

By using

 (389)

we can obtain, for example

 (390)

With the developpement of , we obtain

 (391)

 (392)

 (393)

 (Euler 1748)

The values , gives the formulae

 (394)

 (Grandall 1994)

 (395)

We can also write that which when differentiating gives

 (396)

and obtain pther series such as

By differentiating again , we have

 (400)

etc...

On the same model, Gery Huvent offered what seems an open conjecture, having not read stuff on this subject...

Conjecture 16

 (401)

Example 17

 (402)

Conjecture 18

 (403)

Example 19

 (404)

Conjecture 20

 (405)

Example 21

 (406)

#### 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

 (407)

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

 (408)

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

 (409)

but also

 (410)

and the formula by Comtet dating of 1974 whose proof was tedious...

 (411)

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 [8]

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

 (413)

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

 (414)

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 :

 (415)

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
For
For without and de
Consider, for the path and the integral (with if ).
Since we have

 (417)

If disign a primitive of on the domain containing the path we have

 (418)

The problem hence goes back to the calculation of a primitive !
The null primitive of in is

 (419)

Hence

 (420)

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

 (421)

The primitive of which is nul in is

 (422)

hence

 (423)

In particular,

We can refind that and that

 (425)

And

 (426)

from which

 (427)

Then we have

 (428)

which allows us to find Euler's formula for the dilogarithm

hence

Note 22 We can simplify this equality with Landen's formulae and find the classical result

 (431)

result that we obtain with the developpement into Fourier's series of and Parseval's formula.

in particular

 (Gourevitch)

because

 (432)

This  equality then gives with and a DSE

 (Gourevitch)

We can also establish that

 (433)

We can also calculate the primitive of , we find

This equality allows several applications :

The equality

 (435)

Then with

 (436)

obtained by integration by parts and changing variables because .
We have

 (437)

which gives

 (438)

Finnaly, starting from

 (439)

that we developpe like in the previous example, we find

With and taking into account the value of

 (440)

which gives

 (441)

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

 (Comtet)

This last equality, gives with and the DSE of the equality

 (442)

With the result is less simple, however we have

By combining with 433, we have

 (444)

_

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

 (445)

Proposition 23

 (446)

The previous equality is valid on if
If or the previous equality is valid on

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

which gives

For the folks mathematics We then easily show that

 (459)

The minimum of the trinomial is obtain in , funny !

##### 11.4.3 Uses of

Uses of the primitive for We have for and

 (460)

Hence for or we obtain

 (461)

Starting from

 (462)

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

 (466)

Starting from

 (467)

the change of variabl gives

 (468)

As an application, we have with

With

##### 11.4.4 Uses of

Uses for the primitive of We have for and

 (471)

Starting from

 (472)

by change of variables, we obtain

 (473)

With , we get

It's with this last equality that Apery proved the irrationality of .
With we have

Uses of the primitive of By the same kind of method we obtain

which with gives

then with

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 [9] (4.241), we get the formula

 (484)

Nice formula, putting in evidence the arrival of a constant linked to elliptical integral of first sort, not bad...