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  "body": "\n## Summary\n\nA Horn hypergeometric series is a double power series $\\sum_{m, n} A_{m, n} x^m y^n$ with a special kind of coefficients.\nMoving one step in either index multiplies the coefficient by a rational function of $(m, n)$.\nThis is the two-variable form of the defining property of the Gauss [[hypergeometric series|428M453CR2]], whose consecutive coefficients have a rational ratio.\nHorn classified the convergent series of this type of order two.\nHis list has 34 essentially distinct series, including the four Appell series and their confluent forms.\n\n## Explanations\n\n### L1 human\n\nIn one variable, a hypergeometric series is a power series where each coefficient is the previous one times a fixed rational function of the index.\nIn two variables the coefficients form a grid, and one can step right (in $m$) or up (in $n$).\nHorn's condition asks that each of the two steps multiplies the coefficient by a rational function of the position.\nFor instance $A_{m, n} = \\frac{(m + n)!}{m!\\, n!}$, the coefficients of $\\sum (x + y)^N$, give the ratios $\\frac{m + n + 1}{m + 1}$ and $\\frac{m + n + 1}{n + 1}$.\n",
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