FW: a special 3_F_2

["Daniel Lozier" <lozier@nist.gov>]

Editor's Note:  Sorry for the delay in posting this message.
Dan Lozier

-----Original Message-----
From: Victor Adamchik [mailto:adamchik@ux10.sp.cs.cmu.edu]
Sent: Sunday, April 01, 2001 6:16 PM
To: dlozier@nist.gov; irine_c_p@hotmail.com
Subject: Re: FW: a special 3_F_2


        >But I suspect there is no closed form formulae
        >for this sum for arbitrary b2

Your F32 can be obtained from F21 by (n+1) times integration of

   Gamma[2 + b1 + n]/Gamma[1 + b1]
   z^b1*Hypergeometric2F1[1 + b1, -b2, 1 + b1 - n, z]

wrt z and then setting z->1.  Since there is a closed form for above
2F1, peforming integration one can get a finite sum for F32.

Another and simpler approach is to apply one of the F32(1)
transformation formulas. It will give you the following finite sum:

HypergeometricPFQ[{1+b1,1+b1,-b2},{1+b1-n,2+b1+n},1]
==
Gamma[2+b1+n] Gamma[1+b2+n]/(Gamma[n+1] Gamma[2+b1+b2+n]) *
HypergeometricPFQ[{1+b1,-b2,-n}, {1+b1-n, -b2-n}, 1]

assuming n is a positive integer.

Victor Adamchik