# Appell series F3 on Mathematica

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.

• 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. – J. M.'s torpor Jul 23 '19 at 2:49

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. AppellF3S[{a1_, a2_}, {b1_, b2_}, c_, {x_, y_}, mmax_: 50, nmax_: 50] :=

• 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
• 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