[seqfan] Re: A series for exp(Pi)

[Richard Mathar]

http://list.seqfan.eu/pipermail/seqfan/2009-October/002645.html

jol> Return-Path: <seqfan-bounces at list.seqfan.eu>
jol> From: Jaume Oliver i Lafont <joliverlafont at gmail.com>
jol> To: seqfan at list.seqfan.eu
jol> Subject: [seqfan]  A series for exp(Pi)
jol> 
jol> gp > G(n)=gamma(n)
jol> gp > even(n)=(1+(-1)^n)/G(3*I)/G(-3*I)
jol> gp > odd(n)=3*(1-(-1)^n)/G(1/2+3*I)/G(1/2-3*I)
jol> gp > a(n)=2^(n-1)*G(n/2+3*I)*G(n/2-3*I)*(even(n)+odd(n))
jol> 
jol> Six gamma functions with different argument appear. Is there a simpler solution?

The standard chapters in the Abramowitz-Stegun book give:
Gamma(3i)*Gamma(-3i) = Pi/3/sinh(3*Pi) = 0.0001690...
Gamma(1/2+3i)*Gamma(1/2-3i) = Pi/cosh(3*Pi) = 0.00050705..

Richard Mathar