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  "body": "\n## Definition\n\nLet $a, a', b, b', c, c'$ be complex numbers, with every lower parameter $c, c'$ outside $\\set{0, -1, -2, \\dots}$.\nWrite $(x)_n$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThe four *Appell series* are the double power series in two complex variables $x, y$:\n\n1. $F_1(a; b, b'; c; x, y) = \\sum_{m, n \\ge 0} \\frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\\, m!\\, n!} x^m y^n$, convergent for $\\abs{x} \u003c 1$ and $\\abs{y} \u003c 1$;\n2. $F_2(a; b, b'; c, c'; x, y) = \\sum_{m, n \\ge 0} \\frac{(a)_{m + n} (b)_m (b')_n}{(c)_m (c')_n\\, m!\\, n!} x^m y^n$, convergent for $\\abs{x} + \\abs{y} \u003c 1$;\n3. $F_3(a, a'; b, b'; c; x, y) = \\sum_{m, n \\ge 0} \\frac{(a)_m (a')_n (b)_m (b')_n}{(c)_{m + n}\\, m!\\, n!} x^m y^n$, convergent for $\\abs{x} \u003c 1$ and $\\abs{y} \u003c 1$;\n4. $F_4(a; b; c, c'; x, y) = \\sum_{m, n \\ge 0} \\frac{(a)_{m + n} (b)_{m + n}}{(c)_m (c')_n\\, m!\\, n!} x^m y^n$, convergent for $\\sqrt{\\abs{x}} + \\sqrt{\\abs{y}} \u003c 1$.\n\n## Remarks\n\nSetting $y = 0$ leaves the terms with $n = 0$, and each series becomes a Gauss [[hypergeometric series|428M453CR2]] in $x$; for instance $F_1(a; b, b'; c; x, 0) = {}_2F_1(a, b; c; x)$.\nExample: with $b = c$ and $b' = c'$, the series $F_2$ becomes $\\sum_{m, n} \\frac{(a)_{m + n}}{m!\\, n!} x^m y^n = \\sum_{N \\ge 0} \\frac{(a)_N}{N!} (x + y)^N = (1 - x - y)^{-a}$, by the binomial theorem; this converges when $\\abs{x} + \\abs{y} \u003c 1$.\nNon-example: for $a = 1$ the same series at $x = y = \\frac12$ is $\\sum_N 1$, which diverges; the point lies on the boundary $\\abs{x} + \\abs{y} = 1$ of the region of convergence.\nFor $\\operatorname{Re} c \u003e \\operatorname{Re} a \u003e 0$, with $\\Gamma$ the [[gamma function|0H5SYBSED9]], $F_1$ has Picard's integral representation\n$$F_1(a; b, b'; c; x, y) = \\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(c - a)} \\int_0^1 t^{a - 1} (1 - t)^{c - a - 1} (1 - xt)^{-b} (1 - yt)^{-b'}\\, dt.$$\n",
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