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From: Axel Vogt <&noreply@axelvogt.de>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: A hypergeometric formula
Date: Wed, 10 Sep 2014 20:54:45 +0200
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Xref: csiph.com comp.soft-sys.math.maple:946

On 10.09.2014 20:32, peter.luschny@gmail.com wrote:
>>> 2*GAMMA(3/2)*hypergeom([1/2,0],[3/2,0,-1/2],-1) /sqrt(Pi) = ?
>> (**) y := 2*GAMMA(3/2)*hypergeom([1/2,0],[3/2,0,-1/2],-1) /sqrt(Pi);
>>                     y := hypergeom([1/2], [-1/2, 3/2], -1)
>> (**) simplify(y);
>>                     sin(2) - cos(2)
>
> Thank you Joe!
>
> Now this was the easy part. Next the question: Why does Wolfram Alpha
> (and presumably Mathematica) gives a different answer?
>
>      2 Gamma[3/2] (HypergeometricPFQ[{1/2, 0}, {3/2, 0, -1/2}, -1]/Sqrt[Pi])
>
> Peter

I suggest to consider only the 2F3. Wolfram Alpha says " = 1"
and writes it as series in Pochhamer symbols. That series is
seen to be sin(2) - cos(2) by Maple (which re-writes the very
task as 1F3, dropping the zeros).

Feeding the series to Alpha gives sin(2) - cos(2)

http://www.wolframalpha.com/input/?i=HypergeometricPFQ[{1%2F2%2C+0}%2C+{3%2F2%2C+0%2C+-1%2F2}%2C+-1]

http://www.wolframalpha.com/input/?i=sum%28%28-1%29^k*pochhammer%281%2F2%2Ck%29%2Fk!%2Fpochhammer%28-1%2F2%2Ck%29%2Fpochhammer%283%2F2%2Ck%29%2Ck+%3D+0+..+infinity%29