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  "body": "\n## Summary\n\nThe Appell series $F_1, F_2, F_3, F_4$ are four double power series in two variables $x$ and $y$ that generalise the Gauss [[hypergeometric series|428M453CR2]].\nTheir coefficients are ratios of [[Pochhammer symbols|YZHM3HSM0F]] in the two summation indices.\nEach of them reduces to a Gauss hypergeometric function when one of the variables is $0$.\nThey appear in integrals of products of powers, in the theory of algebraic curves and in physics.\n\n## Explanations\n\n### L1 human\n\nThe Gauss hypergeometric series is a power series in one variable whose coefficients are built from rising factorials.\nTo get a series in two variables, one lets the rising factorials depend on $m$, on $n$, or on $m + n$, where $x^m y^n$ is the monomial.\nThere are four natural ways to do this that give genuinely new functions, and these are Appell's four series.\nFor example $F_1(a; b_1, b_2; c; x, x) = {}_2F_1(a, b_1 + b_2; c; x)$, so on the diagonal $F_1$ collapses back to one variable.\n",
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