[seqfan] Re: Question on 2F1 from Karol A. Penson
Richard J. Mathar
mathar at mpia-hd.mpg.de
Wed Jul 10 12:02:17 CEST 2013
In answer to http://list.seqfan.eu/pipermail/seqfan/2013-July/011394.html:
With the standard integral formula
(see for example equation (9.68) in http://arxiv.org/abs/1207.5845 )
2F1(1,1/2+n; 3+n | -3)
= Gamma(3+n)/Gamma(1/2+n)/Gamma(3+n-1/2-n) *M
where
M = integral_{t=0..1} t^(1/2+n-1)*(1-t)^(3+n-1/2-n-1)/(1+3*t) dt
and for this value M Maple predicts
> t^(n-1/2)*(1-t)^(3/2)/(1+3*t) ;
> int(%,t=0..1) ;
1/2 / GAMMA(n - 1/2) hypergeom([1, -1 - n], [3/2 - n], -1/3)
3/4 Pi |1/3 ------------------------------------------------------
\ GAMMA(2 + n)
(-3 - n) 1/2 \
+ 32 3 Pi sec(Pi n)|
/
and this seems to lead to a terminating 2F1(1,-1-n; 3/2-n; -1/3)
for positive integer n and eventually Gamma-ratios.
Perhaps this is all wrong, but worth looking at.
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