Bug in HypergeometricPFQ[{}, {1/2, 3/4, 3/4}, x]]

• To: mathgroup at smc.vnet.net
• Subject: [mg96055] Bug in HypergeometricPFQ[{}, {1/2, 3/4, 3/4}, x]]
• From: Robert Berger <rberger08 at sinh.us>
• Date: Tue, 3 Feb 2009 07:16:59 -0500 (EST)

```Hello,

it seems that the Linux versions 5.2, 6.0.3 as well as 7.0 (and maybe
also the Windows and OSX versions) of Mathematica exhibit at nasty bug
regarding the evaluation and simplification of HypergeometricPFQ[{},
{1/2, 3/4, 3/4}, x]].

The bug can easily be reproduced by the following test:

g[x_] := f[x] - (9/32)*f'[x] - (69/16)*x*f''[x] - 5*x^2*f'''[x] -
x^3*f''''[x]
f[x_] := HypergeometricPFQ[{}, {1/2, 3/4, 3/4}, x];

By means of an auxiliary calculation, it can be shown that f[x] is one
solution of g[x]==0. Therefore g[x] evaluated at any arbitrary point
should be near zero.

Mathematica 5.2: N[g[1]] is 0.902177 or N[g[0.5]] is 1.40725
Mathematica 7.0: N[g[1]] is 0.902177 or N[g[0.5]] is 1.40725
Mathematica 6.0.3: N[g[1]] is 1.94289*10^-16 or N[g[0.5]] is
1.94289*10^-16

As one can see, Mathematica 6.0.3 yields the correct result.

Also interesting is that if you replace
f[x_] := FunctionExpand[HypergeometricPFQ[{}, {1/2, 3/4, 3/4}, x]];
then Mathematica 6.0.3 yields also the wrong results N[g[1]]=0.902177
and N[g[0.5]]=1.40725!

It seems that the result of FunctionExpand[HypergeometricPFQ[{}, {1/2,
3/4, 3/4}, x]] which is "Sin[4 x^(1/4)]/(8 x^(1/4)) + Sinh[4
x^(1/4)]/(8 x^(1/4))" is not (generally) valid.

Any ideas how to get rid or workaround about that nasty bug?

Kindly regards,
Robert.

```

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