@Mariusz, thanks for finding that. Author Jan Mangdalan is building a prolific reputation for himself. Even good programming can not circumvent difficulties in the theory itself, for example:

AFoxHToMeijerG[
 FoxH[{{{1/2, 2343234/1111111111111}, {1/2, 
     1/4}}, {}}, {{{(1/2)/5, (1/2)/4}}, {{0, 1}}}, z]]

General::nomem: The current computation was aborted because 
there was insufficient memory available to complete the computation.

Throw::sysexc: Uncaught SystemException returned to top level. 
Can be caught with ..., _SystemException].

That's not the original programmer's fault, so don't blame Jan!

I'm not sure if there are still issues here, but if so tickets can be submitted through support page. I think that Computer Calculus will always be a priority at Woflram Research, unfortunately, if the basing is run out of US, given the situation with a new wave of "math reform", especially in California, it may be a low priority (especially relative to, say, what's going on in France and / or Germany).

I'm not sure what else we can do at the moment...

Easy, check if Wolfram's FoxH[] can work for some simple combination of parameters. Trinomial equations are good to start with. Verify the golden ratio case that initiated this discussion or any other similar case.

Mariusz gave a code that works fine using Meijer's G. It is widely acknowledged that Wolfram´s Meijer G implementation works excellent and it is indeed very robust for both symbolics and numerics.

So it would be natural, to solve such situations jumping from FoxH to Meijer's G whenever it is possible, and this occurs always if all scaling parameters are rationals, being this a very frequent case. As we have said, this works just if denominators are not great and irreducible, but what is great or irreducible?, definition depends on computing resources. On the other side there is a huge quantity of common cases, that are not irreducible and software should solve them, for instance this simple golden ratio quadratic equation.

Brad, in my note FoxH.pdf equations H20, H21 & H22 provide the direct general conversion from FoxH to Meijer G.

what else we can do

Pass it to the Development Team to code it inside FoxH. I am absolutely sure that issues like this golden ratio unevaluated and other similar cases will disappear completely.

Have a nice Time,

Jorge

Hi Brad,

I have lifted up a constructive proof in the attached document. It uses Meijer G equivalence for FoxH whenever all scaling parameters are rationals.

Yours Truly

Attachments:

FoxH.pdf

Hi Brad,

Concerning FoxH or Fox-Wright, it is known that just when all parameters $A_j\;$and $B_k\;$ are rationals both functions can be converted always to (most times non-trivial) finite linear combinations of Meijer G and sometimes also to pFq (A,B integers). Since Meijer G and pFq are true hypergeometric functions they have p-recurrences and are diferentiably finite.

For generalized FoxH and $\,_p \Psi_q$ I have read it with some of the reduction formulae to Meijer G, in some of the works and books by Kilbas, Saxena, Mathai, Haubold, Karp, Kiryakova, if I remember.

For trinomial equations reducible to rational exponents $\alpha=\frac{m}{n}$ of this kind $y=x^m+c\,x^n,\;m,n\;$ integers $\,_1 \Psi_1\;$solutions based on pFq finite sums or Meijer G relationships is found in Müller & Moskowitz (1995) paper I uploaded in my previous reply. Look there at Eqs 3.3, 3.5, 3.6 and 3.7 for finite sums of pFq or 5.1 and 5.2 for Meijer G.

A good algorithm strategy for FoxH (and Fox-Wright) is to separate the symbolic case from numerical cases. For irrational parameters and rational parameters A,B having great irreducible denominators as you point out, I think that there is not much to do with symbolics even if you have huge computing resources. Numerics on the other side could be approached in an unified way through quadratures.

For the trinomial equation having A,B irrationals or rationals with great irreducible denominators, a good numeric root finder can solve it.

My Best,

Jorge

Hi Mariusz,

Do you have an opinion whether the Fox Wright or Fox H functions are Differentiably Finite?

I'm tuck on trying to answer this question before rushing on to calculate values according to some theory I don't have time to fully understand.

The code looks promising. Funny example calculating the Golden Ratio by the most complicated means possible! I ran another simple test, and found the implementation does pretty well on integer coefficients:

dat1 = With[{exp = #}, FOXH[
      {{{0, exp}}, {}}, {{{0, 1}}, {{-1, exp - 1}}}, 1]
     ] & /@ Range[2, 6];
Print[]=G,G,G...

dat2 = Flatten[Select[Cases[x /. NSolve[x^(#) + x == 1, x],
      _Real], # > 0 &] & /@ Range[2, 6]];

Union[Chop[dat1 - dat2]]
Out[]={0}

All values computed using Meijer G. For decimals the method was unpredictable and I found at least one suspicious output, probably wrong. There's enough error room in these tests for non-D-Finite hypothesis to remain plausible.

Together with Fox-H, MeijerG, and pFq are the four classical generalized (multi-parameter) univariate hypergeometric functions.

That is what I'm willing to debate, whether or not even Fox-Wright belongs in the hypergeometric class. Refer back to your preferred reference, equations 13.5 and 13.7. The analogy is based on similarity between one particular expansion, not upon fundamental definitions. For example compare the following:

FullSimplify[Times[Gamma[a + n ]/Gamma[b + n],
  Gamma[b + n + 1]/Gamma[a + n + 1]]]
Out[]=(b+n)/(a+n)

FullSimplify[Times[Gamma[a + n A ]/Gamma[b + n B],
  Gamma[b + (n + 1) B]/Gamma[a + (n + 1) A]]]
Out[]=(Gamma[a + A n] Gamma[b + B + B n])/(
Gamma[a + A + A n] Gamma[b + B n])

Is the second expression the result of incomplete programming of FullSimplify? Or does it refuse to simplify to a ratio of $n$-polynomials because such a simplification does not exist? You should try and answer this question by analyzing the coefficient ratio. The term Hypergeometric refers to those deferentially finite functions whose p-recurrences take a most simple (but not necessarily trivial) form. As an academic researcher, you need to understand what that statement means in terms of the calls written above.

However, it looks like you beat me to finding out about inclusion of FoxH. Now here is something else interesting I noticed in theory. Under "neat examples" there is a table of closed forms, all of which I think are Differentiably finite. This table caused me to doubt my earlier conjecture, but maybe not. It could be a case of introducing a simplicity bias. For example, elliptic functions are highly non-linear, but have a harmonic limit. Something similar could be happening here.

Comparing those examples with your posted code, I thought that your list nesting looks correct, but checking the series expansions nothing happened. If you think the series definition will work, it should be easy for you to program yourself. That's what I was getting at in my previous thread.

If you're not worried about getting results right away, you could try to perform some calculus on Mellin-Barnes integrals. However, this is not really recommendable unless you've already mastered a range of easy examples. For example here's a nice demo on relatively easy integrals.

Thanks for sending that paper, it looks really interesting! According to Stephen Wolfram's Blog:

And in Version 13.0 we’re finally done! All the functions in Abramowitz & Stegun are now fully computable in the Wolfram Language.

It looks like you have found a special function, which isn't in A&S. Depending on your constraints for the exponent (as discussed in your source) the Lambert W Function might work. If you have non-integer exponents, then you may need to sum using the Gamma Function, which should also be somewhere in A&S.

The real problem implementing such functions is that, as presented, they may not be well-defined. For example, wikipedia compares your Fox functions with Meijer G and Hypergeometric F,as does the source article. Like newly added Heun G--which solves some of the differential equations in my recent dissertation--Meijer G and Hypergeometric F. are both differentiably finite, meaning that they satisfy (relatively) simple ordinary differential equations. I don't know, but my guess is that your two Fox functions, as being essentially inversive, are more like LambertW, whose defining equation contains a tricky non-linear term.

The Bell polynomials are very interesting from a combinatorial perspective, and powerful technology if utilized correctly. However, most natural scientific researchers ultimately have to decide pragmatically against too much formalism in order to get ahead of schedule on important applications and "neat examples".

Hope that helps.

--Brad