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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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10 Introduction to factorials and combinations

What is very interesting, is that we will always be able to observe the same kind of constant even with the introduction of an equivalent combination or factorial (as long as we stay in a s+1Fs  ), this is why p2  and the logarithms will represent the essential results given by simple hypergeometric. On the other hand, as soon as we introduce a squared combination for example, we will obtain different kind, something to do with the gamma function most of the time (and so is p2   ! but of a less obvious way).

10.1 A first example

The introduction of the combination is justified by several very similar series with or without the presence of combinations. For example,

  2    sum  oo  1      oo  sum    1
p  = 6    n2 = 18   n2Cn--
      n=1        n=1   2n
(Euler)

        oo  sum            sum  oo 
p4 = 90   -14 = 3240    -41n--
       n=1n     17 n=1 n C2n
(Comtet 1974)

       oo  sum  (--1)n-     oo  sum  ---Cn2n---
p = 4   2n + 1 = 2   4n(2n+ 1)
     n=0          n=0
(362)

Which means that in fact as soon as we find a result to this function, we have a little hope to find the same kind of result with a combinations with a central binomial coefficient for example. It's simply due to the form quite close to generators functions of those series (1211).

10.2 Umbral calculus

We can establish a link betweens the series with combinations (k = 1  more precisely) and the series without (k = 0  ) with the help of the acceleration of the convergence by Euler's method. This is a particular case of the formula for "finite differences" which is the discrete version of Taylor's expansion (                  '
f(x+ h) = f(x)+ hf1(x!)+ h2f”(2!x)+ h3f(3)3!(x)+ ...  ). It's the theory known under the name of ”Umbral calculus”, whose origines goes back to Sylvester (1814-1897). The acceleration method by Euler consist to replacing the sum of a serie by the sum of the difference in terms of the serie.

More precisely, we choose an alternating serie  sum o o k=0(-1)kak  and we calculate the finite difference of order k  for a0   :

  k      sum k    m  m
D  a0 =    (- 1) Ck ak-m
       m=0
(363)

then

 oo  sum            oo  sum      k k
  (- 1)kak =    (-1)-D-a0-
k=0          k=0    2k+1
(364)

The trick, it's that the first term is sometime written as combinations or factorial ratio and other symbols of Pochammer ! Application :

We know that 2Y (- 1,1, 1)= 4 sum o o  (-1)k= p
         2       k=0 2k+1  . Hence we have ak = -1--
     2k+1  .

Dka
    0  being quite delicate to calculate directly, we let the function

           sum k         x2(k-m)+1
Dka0(x) =    (- 1)mCmk -----------
         m=0         2(k- m) + 1
(365)

We are going to look for the value in 1  of course. And we calculate it !

pict

where B(x,y) = GG(x()x+Gy(y))  is the complete Bêta function which is worth in this case k > 0  integers B (1,k +1)=  --k!--= 22k+2((k+1)!)2= --22k+2--
   2         (12)k+1    (k+1)(2k+2)!   (k+1)Ck2+k1+2   (interesting...).

From which we obtain

pict

and hence according to the formula 364

pict

Unfortunatly, I have not found how to generalise this method... However, we have still the more general formula

 oo  sum   (- 1)k    1  sum  oo   (k!)
   --------= -    ---(--)-
k=0(q.k + r)  q k=1k2k  rq k
(369)

or even

 oo  sum   (- 1)ixqi    r-1 sum  oo  (- 1)i (r        xqi  )
   qi(q.i+-r) = qq     2i+1-B  q,i+ 1,q--xqi
i=0                 i=0
(370)

where B is an incomplete Bêta , if I'm not mistaken...

Similarly, by using the link between the incomplete bêta and the hypergeometric

pict

where  F
2 1  is the hypergeometric function.

Basicly, the introduction to factorials can be interesting, very interesting....


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