Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg497] Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]*From*: "NELSON M. BLACHMAN" <blachman at gtewd.mtv.gtegsc.com>*Date*: Wed, 01 Mar 1995 22:28:48 PST

This is a postscript to my 26 February message (reproduced below) complaining about Mma's not giving a numerical value for HypergeometricPFQ[{1/2,1,1},{1.45,1.55},1]. I've since found that Mma does evaluate things like HypergeometricPFQ[{1/2,1,1},{1.4,1.6},0.99999], but it takes something like an hour on my 486DX33 PC--and it takes longer and longer as the last argument gets closer and closer to 1. So it's good that Mma quickly announces its inability to compute HypergeometricPFQ[{1/2,1,1},{1.4,1.6},1]. A different method is evidently needed when the last argument is 1--provided that the sum of the components of the middle argument exceeds the sum of the components of the first argument. (If not, the hypergeometric function seems likely to be infinite.) So I continue to hope for a formula for 3F2[{a,b,c},{d,e},1] as a ratio of products of gamma functions of (e, f, and e + f minus 0, a, b, c, a + b, etc.), though I've so far been unable to devise a satisfactory conjecture of this sort. Nelson M. Blachman >From: GTEWD::BLACHMAN "NELSON M. BLACHMAN" 26-FEB-1995 23:54:14.57 >To: MX%"mathgroup at christensen.cybernetics.net" >CC: BLACHMAN >Subj: Numerical Evaluation of HypergeometricPFQ I was pleased to see just now that Mma's able to evaluate Integrate[Sin[t u]/Sin[t],{t,0,Pi/2}] in terms of HypergeomtricPFQ. When I asked it to plot the result, however, I found it apparently unable to determine numerical values for HypergeomtricPFQ. Maple can sometimes compute numerical values for HypergeomtricPFQ, but for the particular denominator indices here it complains of iteration limits' being exceeded. Does anyone know of a way to get Mma 2.2 to evaluate HypergeomtricPFQ numerically? I suspect there's a simple expression for HypergeomtricPFQ[{ },{ },1] in terms of gamma functions when P = Q + 1; there is, anyhow, if P = 2 and Q = 1. Does anyone know if that's true and, if so, what it is? Nelson M. Blachman GTE Government Systems Corp. Mountain View, California Mathematica 2.2 for DOS 387 Copyright 1988-93 Wolfram Research, Inc. In[1]:= Integrate[Sin[t u]/Sin[t],{t,0,Pi/2}] 3 u Pi u 1 3 - u 3 + u (- + -) u Cos[----] HypergeometricPFQ[{-, 1, 1}, {-----, -----}, 1] 2 2 2 2 2 2 Out[1]= ------------------------------------------------------------------- 3 u 1 u 1 u 4 (-(-) - -) (-(-) - -) (- - -) 2 2 2 2 2 2 In[2]:= f[t_]:= %1 /. u -> t Simplify ELIMINATES THE FIRST FACTOR IN THE NUMERATOR AND IN THE DENOMINATOR. In[3]:= f[.1] // N Out[3]= 0.0997665 HypergeometricPFQ[{0.5, 1., 1.}, {1.45, 1.55}, 1.]