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:
https://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/04/06/01/
For specificity, we can just consider the case $q=2$, which is also described in
https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/04/06/01/
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:
In[289]:=
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?