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  "body": "\n## Summary\n\nThe Laguerre polynomials $L_n^{(\\alpha)}$ are the [[classical orthogonal polynomials|47WJ34NC9K]] on the half-line $[0, \\infty)$ for the weight $x^\\alpha e^{-x}$.\nThe case $\\alpha = 0$ gives the ordinary Laguerre polynomials $L_n$.\nThey solve the Laguerre differential equation.\nThey appear in the radial wave functions of the hydrogen atom and in Gaussian quadrature on $[0, \\infty)$.\n\n## Explanations\n\n### L1 human\n\nOn the half-line from $0$ to infinity, the weight $e^{-x}$ gives more importance to small numbers than to large ones.\nThe Laguerre polynomials are the polynomials that are perpendicular to each other for this weight.\nThe first ones are $L_0 = 1$, $L_1 = 1 - x$ and $L_2 = \\frac{x^2 - 4x + 2}{2}$.\nFor instance, $L_1$ is perpendicular to $L_0$ because $\\int_0^\\infty (1 - x) e^{-x} \\, dx = 1 - 1 = 0$.\n",
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