Thank you Mariusz.

I appreciate it. This is very useful for me. I will check it against the code I'm programming making use of the formulae I've found.

Thanks a lot again,

Jorge

.

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,

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

Do you have a constructive, computable proof?

About this question. I think this path provides a constructive proof to transform Fox-H into Meijer G when all $A_j\;$ and $B_k\;$ are rationals

Compute $C = \;$LCM[ All denominators of $A_j\;$,$B_k\;$ ]

Write $A_j\;= n_j / C,\;B_k\;=n_k/C,\;$

Use Legendre multiplication formula for Gamma functions

in the Fox-H Mellin-Barnes integrand $c\;$ is any of $a_j,\;1-a_j,\;b_k,\;1-b_k\;$

$\gamma\;$ is any of $n_j,\;n_k,\;$

$ z\;$ is either $+s/C\;$ or $-s/C\;$ being $s\;$ the integration variable

Change the integration variable to a new one $\sigma=s/C\;$

Check that the integration contour separates numerator poles properly.

Voilá. We have got a Meijer G function. All factors A,B are now +1 or -1

My Best

Jorge

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

I don't know.

I found in WOLFRAM FUNCTION REPOSITORY a very usefull function FoxHToMeijerG.

See here.

Regards.

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.

Hi Brad,

Thanks a lot for your answer, for non-integer exponents a trinomial equation can be solved in terms of confluent Fox-Wright function, that is a simple series generalization of 1F1 hypergeometric function adding two sets more of positive parameters. It plays a similar role as Lambert W to solve trascendent equations having logarithm and linear terms (in this case unknown variable with non-integer -real or complex- power exponents and linear terms) . Fox-Wright $\,_{p}\Psi_{q}$ function is not new, it is a classical function. Together with Fox-H, MeijerG, and pFq are the four classical generalized (multi-parameter) univariate hypergeometric functions.

Now that Mathematica has implemented Fox-H, the generalized Fox Wright $\,_{p}\Psi_{q}$ can also be easily implemented since it is a special case of that function. It is also a "superfunction" but it is much simpler, since , besides a Mellin-Barnes integral, it has also a series form much like pFq, it does not depends on m, n and the confusing splitting brackets that these indices produce on FoxH.

I do not know what happens with FoxH in this particular case but roots of Trinomial equations with non integer (real) exponents are excellent test probes to evaluate the numerical performance of such functions. I posted in StackExchange MO and StackExchange Mathematica Forums math formulae for the root of trinomial equations based on these higher hypergeometric functions informing that Wolfram had implemented FoxH function since v12.3

At last, I would like if you can address this reply to Mr. Tigran Ishkhanyan. Perhaps he can explain the behavior of FoxH in this particular case.

Thanks a lot for your reply,

Jorge