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  "body": "\n## Summary\n\nThe Lauricella series $F_A^{(n)}, F_B^{(n)}, F_C^{(n)}, F_D^{(n)}$ are four multiple power series in $n$ variables that extend the [[Appell series|QG9VDK1HN9]] from two variables to any number of variables.\nFor $n = 2$ they are the Appell series $F_2, F_3, F_4, F_1$, and for $n = 1$ each becomes the Gauss [[hypergeometric series|428M453CR2]].\nThe series $F_D$ is the most used: it has a one-dimensional Euler-type integral and it describes periods of certain algebraic curves.\n\n## Explanations\n\n### L1 human\n\nThe Gauss hypergeometric series has one variable, the Appell series have two, and the Lauricella series allow any number $n$.\nThe coefficient of $x_1^{m_1} \\cdots x_n^{m_n}$ is a ratio of rising factorials that depend either on the single indices $m_i$ or on their total $m_1 + \\cdots + m_n$.\nFor example $F_D$ is what one gets by expanding $\\int_0^1 t^{a - 1} (1 - t)^{c - a - 1} \\prod_i (1 - x_i t)^{-b_i}\\, dt$ in powers of the $x_i$.\nEach factor $(1 - x_i t)^{-b_i}$ contributes one variable.\n",
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