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



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Benoit Cloitre

A Christmas formula for p



 

Benoit Cloitre offers a Christmas formula for p  to us !

Formula

If (x) =  prod n -1(x+ i)
   n    i=0  is Pocchammer's symbol, we obtain this general formula for p  , true for any n  :
            sum  oo      k!          sum n  (- 1)j
p = (-1)n+1    -k(-----1)---- 4    (2j-+-1)
           k=0 2  -n - 2 k+1    j=0

Proof

Let F (n,k) = ----k!----
         2k(- n- 12)k+1   and G(n,k) = 2(2n + 1- 2k)F(n,k)  . It is easy to check the WZ-type relationship :

- (2n + 3)F(n+ 1,k)- (2n+ 3)F (n,k) = G(n+ 1,k)- G(n, k).

Thus, using notations in [Wilf], we obtain

                                 n prod -1
a1(n) = a0(n) = - (2n + 3) ==> A(n) =  (-1) = (- 1)n.
                                 j=0

Moreover, G(j,0) = - 4, F (0,k) = 2k(-k!12)-= - 2 prod kj=k1!2j-1
                          k+1  . The classical Euler's sum  sum o o 
  k=01.3...(k!2k-1) = p2  gives  sum    F (0,k) = - p- 4
  k>0  . Now, finally, from [Wilf], we obtain:

        n sum -1    G(j,0)       1    sum               n sum  oo      k!         n- sum  1(-1)j+1
-p - 4 =    a1(j)A(j +-1)-+ A(n)   F(n,k) = (-1)    2k(-n---1)---+ 4    (2j +-3).
        j=0                   k>0              k=0         2 k+1    j=0

And so,             sum o o                sum n   (- 1)j
p = (- 1)n+1   k=02k(-nk!-12)-- - 4  j=0(2j+1)
                        k+1  as wanted.

References

[Wilf]   H.S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theorical Computer Science 3, 1999, 189-192.


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