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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer, let $x = (x_1, \\dots, x_n) \\in \\C^n$, and let $a, a_i, b, b_i, c, c_i$ be complex numbers with every lower parameter $c, c_i$ outside $\\set{0, -1, -2, \\dots}$.\nWrite $(x)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]], and for a multi-index $m = (m_1, \\dots, m_n) \\in \\N^n$ write $\\abs{m} = m_1 + \\cdots + m_n$, $m! = m_1! \\cdots m_n!$ and $x^m = x_1^{m_1} \\cdots x_n^{m_n}$.\nThe four *Lauricella series* are:\n\n1. $F_A^{(n)}(a; b_1, \\dots, b_n; c_1, \\dots, c_n; x) = \\sum_{m} \\frac{(a)_{\\abs{m}} (b_1)_{m_1} \\cdots (b_n)_{m_n}}{(c_1)_{m_1} \\cdots (c_n)_{m_n}} \\frac{x^m}{m!}$, convergent for $\\abs{x_1} + \\cdots + \\abs{x_n} \u003c 1$;\n2. $F_B^{(n)}(a_1, \\dots, a_n; b_1, \\dots, b_n; c; x) = \\sum_{m} \\frac{(a_1)_{m_1} \\cdots (a_n)_{m_n} (b_1)_{m_1} \\cdots (b_n)_{m_n}}{(c)_{\\abs{m}}} \\frac{x^m}{m!}$, convergent for $\\abs{x_i} \u003c 1$ for all $i$;\n3. $F_C^{(n)}(a; b; c_1, \\dots, c_n; x) = \\sum_{m} \\frac{(a)_{\\abs{m}} (b)_{\\abs{m}}}{(c_1)_{m_1} \\cdots (c_n)_{m_n}} \\frac{x^m}{m!}$, convergent for $\\sqrt{\\abs{x_1}} + \\cdots + \\sqrt{\\abs{x_n}} \u003c 1$;\n4. $F_D^{(n)}(a; b_1, \\dots, b_n; c; x) = \\sum_{m} \\frac{(a)_{\\abs{m}} (b_1)_{m_1} \\cdots (b_n)_{m_n}}{(c)_{\\abs{m}}} \\frac{x^m}{m!}$, convergent for $\\abs{x_i} \u003c 1$ for all $i$.\n\nEach sum runs over all $m \\in \\N^n$.\n\n## Remarks\n\nFor $n = 2$ the four series are the [[Appell series|QG9VDK1HN9]]: $F_A = F_2$, $F_B = F_3$, $F_C = F_4$ and $F_D = F_1$.\nExample: for $n = 1$ all four series reduce to $\\sum_{k} \\frac{(a)_k (b)_k}{(c)_k} \\frac{x^k}{k!} = {}_2F_1(a, b; c; x)$, with $b = b_1$ and $c = c_1$ as needed; the check is a direct comparison of coefficients.\nNon-example: the point $x_1 = x_2 = \\frac12$ is outside the region of convergence of $F_A^{(2)}$; with $a = 1$ and $b_i = c_i$ the series is $\\sum_{N \\ge 0} (x_1 + x_2)^N$, which diverges there.\nFor $\\operatorname{Re} c \u003e \\operatorname{Re} a \u003e 0$, with $\\Gamma$ the [[gamma function|0H5SYBSED9]], the series $F_D$ has the integral form\n$$F_D^{(n)}(a; b_1, \\dots, b_n; c; x) = \\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(c - a)} \\int_0^1 t^{a - 1} (1 - t)^{c - a - 1} \\prod_{i = 1}^{n} (1 - x_i t)^{-b_i}\\, dt.$$\n",
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