I am trying to make Mathematica simplify the following expression below.

 HypergeometricPFQ[{-(1/3), 1/3, -(1/2) - m, -m}, {1/2, 1/3 - m, 
   2/3 - m}, 1]
 , Assumptions -> Element[m, Integers] && m >= 1]

This results in some error, complaining about Gamma[-3m]being ComplexInfinity.

It seems that this should always be -1/2, for any positive integer value of $m$. If there is some general formula which can evaluate the above, I would be happy for that too.

  • 1
    $\begingroup$ Table[HypergeometricPFQ[{-(1/3), 1/3, -(1/2) - m, -m}, {1/2, 1/3 - m, 2/3 - m}, 1], {m, 10^3}] seems pretty evidently true. $\endgroup$
    – Roman
    Jul 13 at 19:51
  • $\begingroup$ @Roman Right - which is why I would like to prove it. If Mathematica can understand it, it's likely a special case of some more general formula. $\endgroup$ Jul 13 at 20:17
  • 1
    $\begingroup$ Have you looked through the formulas for $_{q+1}F_q(z=1)$? There seem to be lots of known special cases; maybe some of them aren't in Mathematica's database. $\endgroup$
    – Roman
    Jul 13 at 20:24

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