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I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as $$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_1)_m(a_2)_n(b_1)_m(b_2)_n}{(c)_{m+n}m!\,n!}x^my^n.$$ I wonder if there exist something on Mathematica to represent this function,as it happens for Appell series F1 by means of the command AppellF1, or if one has to use e.g. its double integral representation in order to evaluate it for specific values. I've searched for it but I was unable to find anything useful.

ARWarrior
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1 Answers1

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Please see this link

Weisstein, Eric W. "Appell Hypergeometric Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AppellHypergeometricFunction.html

Your series is defined there, along with some Mathematica code (a link to a notebook) to compute it.

enter image description here

From the linked notebook file:

AppellF3S[{a1_, a2_}, {b1_, b2_}, c_, {x_, y_}, mmax_: 50, nmax_: 50] :=
  Sum[
   ((Pochhammer[a1,m]*Pochhammer[a2,n]*Pochhammer[b1,m]*Pochhammer[b2, n])/
           (m!*n!*Pochhammer[c, m + n])) 
             *x^m*y^n, {m, 0, mmax},{n, 0, nmax}
     ]
mjw
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  • I only see code for $F_1$ (as OP already noted), not for $F_3$. Am I missing something? – AccidentalFourierTransform Jul 10 '19 at 18:06
  • It is in the linked notebook file. – mjw Jul 10 '19 at 18:12
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    ... that's... not very good code if I'm being honest. It's a massive finite sum, useless both for symbolic and numeric computations. – AccidentalFourierTransform Jul 10 '19 at 18:30
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    Well then, I am glad that I posted the code (which I did not write). Now you can improve on it! – mjw Jul 10 '19 at 18:47
  • By the way, the definition of the function is a "massive sum", how else would you compute it? – mjw Jul 10 '19 at 18:59
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    The definition is an infinite sum, which is quite different from a finite sum with $50^2$ terms. That's what I mean by massive: if you evaluate it for non-float parameters, you'll get a massive expression, impossible to work with. Also, OP was asking about built-in methods, they already know about alternative representations. That's why I feel your expression is not particularly useful: it is useless for symbolic manipulations, it is not well suited for numeric computations, and it does not correspond to what OP wanted. Anyway, I hope you don't take this criticism personally... – AccidentalFourierTransform Jul 10 '19 at 19:54
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    So your criticism is "we all know how it is defined in terms of an infinite sum of Pochhammer functions. Pointing to an expression where a finite approximation is 'coded' is not helpful, because any of us could have written the same code in less than a minute. Besides, we want alternative expressions, such as integral representations, or whatever." Okay, I get it. Thanks! – mjw Jul 10 '19 at 20:36
  • AppellF3 is now part of 13.3 version. – Валерий Заподовников Jul 01 '23 at 05:03