I am mucking about with hypergeometric functions and looking at some of the Wolfram identities. In the documentation I found this


It specifically refers to a function written


and claims that its input in mathematica is given by (for some parameters)

HypergeometricPFQ[{{1 + a, 1 + b}, {1}, {1, b}}, {{2, 1 + c}, {}, {1 + b}}, z, z]

I cannot find the definition of such a function anywhere. Moreover, when I attempt to input said function into Mathematica (version 12) it doesn't recognise the input giving error

HypergeometricPFQ: HypergeometricPFQ called with 4 arguments; 3 arguments are expected.

Is this just a deprecated function lying around on the web? Even if so, what is it?

Granted the page is fairly old, but it's still bad form to have that up on the web. And fairly mind-boggling to me not to explain, even just mathematically, what this mystery function is...


To be clear: the function is not the usual HypergeometricPFQ function which I am well aware of and is well documented. It is a unspecified function that appears in past versions of Mathematica to have overloaded the HypergeometricPFQ symbolic function name.

  • $\begingroup$ In the Details section of the documentation the definition is given. $\endgroup$ – JimB Apr 10 '20 at 4:20
  • $\begingroup$ @JimB What? Ctrl+F on that page for "Details" draws a blank... $\endgroup$ – user3353819 Apr 10 '20 at 4:22
  • $\begingroup$ That page you linked to is dated 2001-10-29. That's pretty old documentation! You'll no doubt get better results using more recent documentation, as for the HypergeometricPFQ function. $\endgroup$ – bill s Apr 10 '20 at 4:51
  • $\begingroup$ @bills I appreciate that, but mathematical identities don't go out of date! The function is something, but there is no equivalent listed in the new documentation. That specific idiom (giving HypergeometricPFQ 4 inputs - some of which are themselves lists of lists) is now not valid. Forget, the mathematica aspect - what is that function? Where in the modern documentation is that function described? I am at a loss. $\endgroup$ – user3353819 Apr 10 '20 at 4:55
  • $\begingroup$ Mathematical identities may not go out of date, but software implementations do, as evidenced by this question! But maybe you'll be lucky and someone from Wolfram will remember how they implemented this 20 years ago. $\endgroup$ – bill s Apr 10 '20 at 5:06

Extended comment: In the Details section of HypergeometricPFQ for Mathematica 12.1 is

Definition of HypergeometricPFQ

You seem to think that the right-hand side of the following equation

Symbolic definition

represents Mathematica code (with the two z's). It is not and never was.

  • $\begingroup$ Please read the question properly. The function in question is not ``HypergeometricPFQ'', defined as you have written. It is written differently in mathematical form (see the form $F^{abc}_{def}$). The documentation I have linked to uses the symbol HypergeometricFPQ but has a different number of inputs. Clearly in some old deprecated version of Mathematica whatever function it is overloaded the same function name with a different number of inputs. $\endgroup$ – user3353819 Apr 10 '20 at 4:29
  • $\begingroup$ I have read it properly. You say "HypergeometricPFQ[{{1 + a, 1 + b}, {1}, {1, b}}, {{2, 1 + c}, {}, {1 + b}}, z, z] I cannot find the definition of such a function anywhere. Moreover, when I attempt to input said function into Mathematica (version 12) it doesn't recognise the input giving error". I simply showed you what the documentation says that you say you can't find. Also, Mathematica is telling you that you gave it 4 arguments when it only takes 3. It means what it says. $\endgroup$ – JimB Apr 10 '20 at 5:01
  • $\begingroup$ There is an explicit description of said function $2^{212}_{201}$ and its input is explicitly described in Mathematica code on that webpage. I understand it must be deprecated. Unlike what your edit implies, I don't think it is Mathematica code, I'm just directly followed the webpage. I know its not valid current Mathematica code - I said as much in my original post. So: given that you have now written the function I am asking about: what is that function on the r.h.s.? How do I input it in Mathematica? That page does say to use HypergeometricPFQ. Look at the section "input form" $\endgroup$ – user3353819 Apr 10 '20 at 5:08
  • $\begingroup$ It appears to be a two-variable hypergeometric function somewhat related to AppellF1. $\endgroup$ – JimB Apr 10 '20 at 5:17
  • $\begingroup$ I didn't "write" the function. I just copied the picture from the link you provided. $\endgroup$ – JimB Apr 10 '20 at 5:37

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