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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer.\nThe *Bessel polynomial* of degree $n$ is the [[polynomial|1G0B73WPX8]]\n$$y_n(x) = \\sum_{k=0}^{n} \\frac{(n + k)!}{(n - k)! \\, k!} \\left( \\frac{x}{2} \\right)^k.$$\nThe *reverse Bessel polynomial* is $\\theta_n(x) = x^n y_n(1/x)$.\n\n## Remarks\n\nExamples: $y_0 = 1$, $y_1 = 1 + x$ and $y_2 = 1 + 3x + 3x^2$.\nCheck for the top coefficient of $y_2$: $\\frac{4!}{0! \\, 2!} \\cdot \\frac14 = 3$.\nEach $y_n$ solves $x^2 y'' + (2x + 2) y' - n(n + 1) y = 0$; for $y_1$ this reads $0 + (2x + 2) - 2(1 + x) = 0$.\nNon-example: $1 + 2x$ has degree $1$ but is not a solution for $n = 1$, since $(2x + 2) \\cdot 2 - 2(1 + 2x) = 2 \\ne 0$.\nFor $x \u003e 0$ they are linked to the modified Bessel function of the second kind: $y_n(x) = \\sqrt{\\frac{2}{\\pi x}} \\, e^{1/x} K_{n + 1/2}(1/x)$.\n",
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