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

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg502] Re: [mg497] Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]*From*: bob Hanlon <hanlon at pafosu2.hq.af.mil>*Date*: Fri, 3 Mar 1995 06:10:55

Take a look at "Higher Transcendental Functions, Volume I", Erdelyi, A. et al., McGraw-Hill, 1953; section 4.4, Generalized hypergeometric series with unit argument in the case p = q + 1, pp. 188-190. It provides a few results. Bob Hanlon hanlon at pafosu2.hq.af.mil On Wed, 1 Mar 1995, NELSON M. BLACHMAN wrote: > 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.] > >