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Re: HypergeometricPFQ - simplification

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89317] Re: HypergeometricPFQ - simplification
  • From: sashap <pavlyk at gmail.com>
  • Date: Thu, 5 Jun 2008 00:44:29 -0400 (EDT)
  • References: <g1gqir$1fn$1@smc.vnet.net>

On May 27, 2:16 pm, janos <janostothmeis... at gmail.com> wrote:
> Could anyone show using Mathematica that the expression
>
> -2/27 + (253*HypergeometricPFQ[{-1/2, -1/6, 1/6}, {4/3, 5/3}, -1/729])/
> 108 - (911*HypergeometricPFQ[{1/2, 5/6, 7/6}, {7/3, 8/3}, -1/729])/
> 6298560 +
>  (73*HypergeometricPFQ[{3/2, 11/6, 13/6}, {10/3, 11/3}, -1/729])/
> 9795520512
>
> is equal to
>
> (-2*(-1 + 10*Sqrt[10]))/27? (Using, say, TraceInternal, to see what is
> going on.)
>
> Thank you,
>
> Janos

Dear Janos,

A way to prove this in Mathematica  takes couple of lines of code:

In[1]:= $Version

Out[1]= "6.0 for Microsoft Windows (32-bit) (March 13, 2008)"

(* your expression *)
In[2]:= expr = -(2/27) +
   253/108 HypergeometricPFQ[{-(1/2), -(1/6), 1/6}, {4/3, 5/3}, -(1/
      729)] - (
   911 HypergeometricPFQ[{1/2, 5/6, 7/6}, {7/3, 8/3}, -(1/729)])/
   6298560 + (
   73 HypergeometricPFQ[{3/2, 11/6, 13/6}, {10/3, 11/3}, -(1/729)])/
   9795520512;

Now define

In[3]:= e1 = ({D[
      HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z],
     D[HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z, z],
     D[HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z, z, z]} /. {a -> 0, z -> 1/9} // Expand)

Out[3]= {(35/144)*
  HypergeometricPFQ[{-(1/2), -(1/6), 1/6}, {4/3, 5/3}, -(1/729)],
   (35/8)*
   HypergeometricPFQ[{-(1/2), -(1/6), 1/6}, {4/3,
     5/3}, -(1/729)] - (7*
     HypergeometricPFQ[{1/2, 5/6, 7/6}, {7/3, 8/3}, -(1/729)])/
       124416, (315/8)*
   HypergeometricPFQ[{-(1/2), -(1/6), 1/6}, {4/3, 5/3}, -(1/729)] -
     (7*HypergeometricPFQ[{1/2, 5/6, 7/6}, {7/3, 8/3}, -(1/729)])/
   2304 +
     (35*HypergeometricPFQ[{3/2, 11/6, 13/6}, {10/3,
       11/3}, -(1/729)])/214990848}

Now repeat the same with variable a set to 0 from the outset:

In[4]:= e2 =
 FullSimplify[
  With[{a = 0}, {D[
      HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z],
     D[HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z, z],
     D[HypergeometricPFQ[{-(1/2), -(1/6), 1/6 + a} - 1, {4/3, 5/3} -
        1, -z^3], z, z, z]}] /. z -> 1/9]

Out[4]= {Sqrt[
 11236809 + 778034 Sqrt[73] -
  2000 Sqrt[12368090 + 7780340 Sqrt[73]]]/1458,
 7/324 Sqrt[
  21873 + 10658 Sqrt[73] - 200 Sqrt[-781270 + 106580 Sqrt[73]]],
 35/36 Sqrt[73/2 (15 + Sqrt[73]) + 10 Sqrt[5 (595 + 73 Sqrt[73])]]}

Now solve for hypergeometric functions:

In[7]:= ru =
  Thread[Union[Cases[e1, _HypergeometricPFQ, Infinity]] -> {h1, h2,
     h3}];

and substitute back into your original expression:

In[8]:= FullSimplify[
 expr /. (Solve[((e1 - e2) /. ru) == 0, {h1, h2, h3}] /. (Reverse /@
      ru))]

Out[8]= {2/27 (-1 + 10 Sqrt[10])}

Hope this helps,
Oleksandr Pavlyk
Wolfram Research


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