# Branch cut of generalized hypergeometric function (error in Mathematica documentation?)

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:= (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= 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:=
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= 0. + 1.81299 I


so is the formula in the mathematica documentation wrong?

• Could you add an example of any discrepancies in code with your expected result as well? – MarcoB Jun 8 '20 at 14:04
• 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 $q=1,2$.) – esches Jun 8 '20 at 14:55
• 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. – esches Jun 8 '20 at 15:03
• i added a concrete example of the branch cut i expect for the $q=2$ case, which contradicts the formula from the mathematica documentation – esches Jun 8 '20 at 16:12