Une formule de Noël pour Pi

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Benoit Cloitre

A Christmas formula for $p$

Benoit Cloitre offers a Christmas formula for $p$ to us !

Formula
Proof
References

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.