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.

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    $\begingroup$ Mathematica does not support this function, as you've already been told. If you need to evaluate this numerically, you can either use a number of integral representations, or sum the double series using techniques like in this answer. $\endgroup$ Jul 23, 2019 at 2:49
  • $\begingroup$ Mathematica actually does support is as of version 13.3 $\endgroup$ Jul 3, 2023 at 18:24

1 Answer 1


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] :=
   ((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}
  • $\begingroup$ I only see code for $F_1$ (as OP already noted), not for $F_3$. Am I missing something? $\endgroup$ Jul 10, 2019 at 18:06
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    $\begingroup$ ... that's... not very good code if I'm being honest. It's a massive finite sum, useless both for symbolic and numeric computations. $\endgroup$ Jul 10, 2019 at 18:30
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    $\begingroup$ Well then, I am glad that I posted the code (which I did not write). Now you can improve on it! $\endgroup$
    – mjw
    Jul 10, 2019 at 18:47
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    $\begingroup$ 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... $\endgroup$ Jul 10, 2019 at 19:54
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    $\begingroup$ 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! $\endgroup$
    – mjw
    Jul 10, 2019 at 20:36

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