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  "body": "\n## Definition\n\nLet $(A_{m, n})_{m, n \\ge 0}$ be complex numbers with $A_{0, 0} = 1$.\nConsider the [[formal power series|033WGN00BF]] $\\sum_{m, n \\ge 0} A_{m, n} x^m y^n$.\nIt is a *Horn hypergeometric series* if there are rational functions $f$ and $g$ of two variables such that for all $m, n \\ge 0$:\n\n1. $A_{m + 1, n} = f(m, n)\\, A_{m, n}$;\n2. $A_{m, n + 1} = g(m, n)\\, A_{m, n}$.\n\nWrite $f = P / Q$ and $g = R / S$ with polynomials $P, Q, R, S$ in lowest terms.\nThe *order* of the series is the largest of the degrees of $P, Q, R, S$.\n\n## Remarks\n\nThe two conditions mean that the ratios of neighbouring coefficients are rational in $(m, n)$, as for the Gauss [[hypergeometric series|428M453CR2]] in one variable.\nExample: Appell's $F_1(a; b, b'; c; x, y)$ has $A_{m, n} = \\frac{(a)_{m + n} (b)_m (b')_n}{(c)_{m + n}\\, m!\\, n!}$.\nThen $f(m, n) = \\frac{(a + m + n)(b + m)}{(c + m + n)(m + 1)}$ and $g(m, n) = \\frac{(a + m + n)(b' + n)}{(c + m + n)(n + 1)}$, both rational, so $F_1$ is a Horn series of order $2$.\nNon-example: $A_{m, n} = 2^{m^2}$ gives $A_{m + 1, n} / A_{m, n} = 2^{2m + 1}$, which is not a rational function of $m$, so the first clause fails.\nHorn's list contains $34$ essentially distinct convergent series of order $2$: $14$ complete ones, among them the four Appell series, and $20$ confluent ones.\n",
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