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  "body": "\n## Example\n\nLet $\\lambda \\in \\C$. The *Laguerre equation* is\n$$z\\, y'' + (1 - z)\\, y' + \\lambda\\, y = 0.$$\nIn normalised form, $p(z) = \\frac{1 - z}{z}$ and $q(z) = \\frac{\\lambda}{z}$.\nThen $z p(z) = 1 - z$ and $z^2 q(z) = \\lambda z$ are analytic at $0$, so $0$ is a [[regular singular point|H96HXG6CA0]].\nThe indicial equation is $r (r - 1) + r = r^2 = 0$, a double root $r = 0$.\nBy [[Fuchs' theorem|6EZ400WM94]], there is one solution $y_1 = \\sum_{k \\ge 0} a_k z^k$, analytic at $0$, and a second solution containing $y_1 \\log z$.\nInserting the series gives $(k + 1)^2 a_{k+1} = (k - \\lambda)\\, a_k$.\nIf $\\lambda = n$ is a nonnegative integer, the series stops and $y_1$ is a multiple of the Laguerre polynomial $L_n$ of degree $n$.\n\n## Remarks\n\nFor $n = 1$, $L_1(z) = 1 - z$: then $y' = -1$ and $y'' = 0$, and $z \\cdot 0 + (1 - z)(-1) + (1 - z) = 0$.\nAt $\\infty$, the function $z^2 q(z) = \\lambda z$ is unbounded, so $\\infty$ is an irregular singular point and the equation is not Fuchsian.\nThe generalized Laguerre polynomials $L_n^{(\\alpha)}$, solutions of $z y'' + (\\alpha + 1 - z) y' + n y = 0$, appear in the radial part of the wave functions of the hydrogen atom.\n\n## Explanations\n\n### L1 human\n\nThe Laguerre equation has one tame bad point at $0$, where both exponents are equal to $0$.\nSo one solution is an ordinary power series, and the other one needs a logarithm.\nFor whole numbers $\\lambda = n$, the series solution is a polynomial, the Laguerre polynomial.\n",
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