[seqfan] Re: Question on 2F1 from Karol A. Penson

[Karol]

Dear Richard, thanks for this very useful hint.
   I will have a look at that.

    Best regards, Karol A. Penson



Le 10/07/13 12:02, Richard J. Mathar a écrit :
> 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.
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/