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  "body": "\n## Summary\n\nA confluent hypergeometric function is a solution of [[Kummer's equation|NT917KFR57]] $z w'' + (b - z) w' - a w = 0$.\nThe main one is Kummer's function $M(a, b, z) = {}_1F_1(a; b; z)$, the [[generalized hypergeometric series|VDBAW9PN88]] with one upper and one lower parameter.\nIt is an [[entire function|0G4F0XB9X2]] of $z$.\nIts special cases include the exponential function, the [[Laguerre polynomials|WAMRHWW82R]], the [[error function|7Y56QCZSZJ]] and the [[incomplete gamma function|YB7ZV0HG7K]].\n\n## Explanations\n\n### L1 human\n\nKummer's function is a power series in which each coefficient is obtained from the previous one by a simple rational factor.\nThe coefficient of $z^n$ is $\\frac{a (a + 1) \\cdots (a + n - 1)}{b (b + 1) \\cdots (b + n - 1)} \\cdot \\frac{1}{n!}$.\nWhen $a = b$ the two products cancel and the series is $\\sum z^n / n! = e^z$.\nSo Kummer's function is a two-parameter deformation of the exponential function.\nIt appears whenever a physical problem leads to a second order equation with one regular and one irregular singular point, as for the hydrogen atom.\n",
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