Converting a 3F2 to elementary functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg58157] Converting a 3F2 to elementary functions*From*: carlos at colorado.edu*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)