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  "body": "\n## Definition\n\nLet $a, b, c \\in \\C$.\nThe *hypergeometric differential equation* is the second-order linear [[ordinary differential equation|0NT64JV96A]] for a function $w$ of a complex variable $z$:\n$$z(1 - z) \\, \\frac{d^2 w}{dz^2} + \\big[c - (a + b + 1) z\\big] \\frac{dw}{dz} - ab \\, w = 0.$$\n\n## Remarks\n\nIts singular points on the Riemann sphere are $0$, $1$ and $\\infty$, and all three are [[regular singular points|H96HXG6CA0]].\nThe exponents are $\\set{0, 1 - c}$ at $0$, $\\set{0, c - a - b}$ at $1$ and $\\set{a, b}$ at $\\infty$.\n\nExample: if $c \\notin \\set{0, -1, -2, \\ldots}$, the [[hypergeometric function|463X5V31D9]] $w = {}_2F_1(a, b; c; z)$ is a solution.\nTo check it, put $w = \\sum_n t_n z^n$; the equation gives $(n+1)(n+c)\\, t_{n+1} = (n+a)(n+b)\\, t_n$, which is exactly the recursion of the coefficients $t_n = \\frac{(a)_n (b)_n}{(c)_n n!}$.\nIf moreover $c \\notin \\Z$, a second, independent solution near $0$ is $z^{1-c} \\, {}_2F_1(a - c + 1, b - c + 1; 2 - c; z)$.\n\nNon-example: [[Kummer's equation|NT917KFR57]] $z w'' + (c - z) w' - a w = 0$ is not of this form; it has a regular singular point at $0$, but its singular point at $\\infty$ is irregular.\n\nIn the notation of the Riemann P-symbol, the solutions form\n$$P\\begin{Bmatrix} 0 \u0026 1 \u0026 \\infty \u0026 \\\\ 0 \u0026 0 \u0026 a \u0026 z \\\\ 1 - c \u0026 c - a - b \u0026 b \u0026 \\end{Bmatrix}.$$\n",
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