# Bug in HypergeometricPFQRegularized?

Bug persisting through 11.2

I'am facing a trouble with HypergeometricPFQRegularized. Mathematica hangs if imaginary part of one of the arguments has zero imaginary part with decimal point as the following example shows:

\[Nu]j = 2 + I*0
HypergeometricPFQRegularized[{1/2, 1}, {1 - \[Nu]j,
1 + \[Nu]j}, -14.28]
\[Nu]j = 2 + I*0.
HypergeometricPFQRegularized[{1/2, 1}, {1 - \[Nu]j,
1 + \[Nu]j}, -14.28]

(*
Out[404]= 2

Out[405]= 0.112647

Out[406]= 2. + 0. I

During evaluation of In[404]:= Divide::infy: Infinite expression 3/0 encountered.

During evaluation of In[404]:= Divide::infy: Infinite expression 3/0 encountered.

During evaluation of In[404]:= Divide::infy: Infinite expression 3/0 encountered.

During evaluation of In[404]:= General::stop: Further output of Divide::infy will be suppressed during this calculation.
*)


This code is used in heavy serial calculations which hangs sometimes and my investigation indicated on the above trouble with HypergeometricPFQRegularized. Could someone propose simple workaround? I need effective code that would round [Nu]j when it is close to an integer.

• Perhaps Rationalize or Chop will solve your problem. – Bob Hanlon Feb 8 '18 at 13:30

Diagnosis

Upon inspecting the source code for HypergeometricPFQRegularized with

<< GeneralUtilities
PrintDefinitions[HypergeometricPFQRegularized]


I have tracked that error in the OP's example down to a poorly constructed pattern for the 10th DownValue:

HypergeometricPFQRegularized[a_List,{b1___,Except[_Complex, c_],b2___},z_] := ...


Clearly the programmer intended for the rule to be applied in case c was real, but it fails to match if c is an approximate real number with head Complex.

Furthermore, the RHS of the rule, additionally imposes a Condition:

... := Condition[(*body*), InternalRealValuedNumericQ[c] &&
Block[{rc = Round[c]}, rc <= 0 && c == rc]]


which among other things again checks for the real-valuedness of c, but failed to pass if c has head Complex.

Resolution

To fix this code, I have taken the liberty of modifying that rule (along with parts of the body) so that it functions as it was (likely) intended. From a fresh kernel, run the following:

HypergeometricPFQRegularized[{}, {}, 0]; (*loads definitions*)
Unprotect[HypergeometricPFQRegularized];

(*Modifies the 10th rule*)
DownValues[HypergeometricPFQRegularized] =
ReplacePart[DownValues[HypergeometricPFQRegularized], 10 ->
HoldPattern[HypergeometricPFQRegularized[a_List, {b1___, c_, b2___}, z_] /; PossibleZeroQ[Im[c]]] :>
Module[{temp, v},
temp = Select[{b1, c, b2}, (Last[InternalTestIntegerQ[#]] && # <= 0) &];
v = 1 - Round[Min[temp]];
z^v (Times @@ Pochhammer[a, v] HypergeometricPFQRegularized[a + v,
Append[SystemHypergeometricPFQDumpreduce[{b1, c, b2}, {1 - v}] + v, 1 + v], z])
] /; Block[{rc = Round[c]}, rc <= 0 && c == rc]];

Protect[HypergeometricPFQRegularized];


And now, the numerical evaluation proceeds without problems:

νj = 2 + I*0.;
HypergeometricPFQRegularized[{1/2, 1}, {1 - νj, 1 + νj}, -14.28]

(*  0.112647  *)

• PrintDefinitions` seems like a really useful function! – chris Feb 9 '18 at 7:48