Converting a 3F2 to elementary functions

• To: mathgroup at smc.vnet.net
• Subject: [mg58157] Converting a 3F2 to elementary functions
• Date: Mon, 20 Jun 2005 05:21:34 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In a fluid-mechanics application I need the 3F2 hypergeometric
function (parameter m is a positive integer)

g32[m_,x_]:= x^2*HypergeometricPFQ[{1,2*m-1,1},{m+1/2,2},x^2];

For small m this converts (via FunctionExpand and Simplify) to
elementary functions:

g32[1,x]=   ArcSin[x]^2
g32[2,x]=   (3*(x*Sqrt[1-x^2]+(-1+2*x^2)*ArcSin[x]))/(4*x*Sqrt[1-x^2])
g32[3,x]=   (-5*(x*Sqrt[1-x^2]*(-3-20*x^2+20*x^4)+3*
(1+6*x^2-24*x^4+16*x^6)*ArcSin[x]))/(192*x^3*(1-x^2)^(3/2))
g32[4,x]=   (7*(x*Sqrt[1-x^2]*(45+180*x^2+1324*x^4-3008*x^6+1504*x^8)+
15*(-3-10*x^2-80*x^4+480*x^6-640*x^8+256*x^10)*ArcSin[x]))/
(23040*x^5*(1-x^2)^(5/2))

The conversion time grows quickly, however, as m increases:
m=1: 0.05 sec, m=2: 0.92 sec, m=3: 22.7 sec, m=4: 35 min
so for m=5 I estimate several hours.

Question: would there be a faster built-in way? (I would like
to go up to m ~ 12)

```

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