I am trying to compute the discontinuity around $x=1$ (equivalently, the branch cut) of generalized Hypergeometric functions ${}_{q+1}F_q(a_1,\dots,a_{q+1};b_1,\dots,b_q;x)$. Two formulae are given in the Mathematica documentation:


For specificity, we can just consider the case $q=2$, which is also described in


in the $q=2$ case, this page only gives a formula that is valid when $a_i-a_j$ is not an integer, which is analogous to the last formula in the general $q$ page. I am interested in the general $q$ case, in which case the second to last formula on that page claims to hold for general $a_i-a_j$. Let me evaluate it for one of the cases I am interested in:

In[277]:= (Product[Gamma[Subscript[b, k]], {k, 1, q}]/
         Product[Gamma[Subscript[a, k]], {k, 1, 
           q + 1}]) MeijerG[{{1}, {Subscript[b, 1], 
          Subscript[b, q]}}, {{Subscript[a, 1], Subscript[a, 2], 
          Subscript[a, q + 1]}, {}}, E^(Pi I) (1/x)] /. q -> 2 /. 
     Subscript[a, 1] -> 3/2 /. Subscript[a, 2] -> 2 /. 
   Subscript[a, 3] -> 5/2 /. Subscript[b, 1] -> 1 /. 
 Subscript[b, 2] -> 3

Out[277]= HypergeometricPFQ[{3/2, 2, 5/2}, {1, 3}, x]

as you see, this formula has given a smooth limit for $x+i \epsilon$, i.e. it shows no branch cut.

But this cannot be true, bc we can see that their is a branch cut by simple numerical evaluation:

HypergeometricPFQ[{3/2, 2, 5/2}, {1, 3}, 2 + I 10^-10] - 
  HypergeometricPFQ[{3/2, 2, 5/2}, {1, 3}, 2 - I 10^-10] // N

Out[289]= 0. + 1.81299 I

so is the formula in the mathematica documentation wrong?

  • 1
    $\begingroup$ Could you add an example of any discrepancies in code with your expected result as well? $\endgroup$ – MarcoB Jun 8 '20 at 14:04
  • $\begingroup$ thanks for the comment. the mismatch between the two formulae was my typo, but I am still confused by the implication of these two formula (and the comparison between the general $q$ case and specific values of $q$, such as $q=1,2$.) $\endgroup$ – esches Jun 8 '20 at 14:55
  • $\begingroup$ whats confusing is that the second to last formula in functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/… seems to basically just be the definition of the hypergeometric function, as written in terms of the MeijerG function. bc even for non-integer $a_i-a_j$, i find that this formula says that there is no branch cut, which contradicts the last formula on that page that says that there definitely is a branch cut. $\endgroup$ – esches Jun 8 '20 at 15:03
  • $\begingroup$ i added a concrete example of the branch cut i expect for the $q=2$ case, which contradicts the formula from the mathematica documentation $\endgroup$ – esches Jun 8 '20 at 16:12

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