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  "body": "\n## Summary\n\nThe Charlier polynomials $C_n(x; a)$ are the orthogonal polynomials of the [[Poisson distribution|0XB9GQTN9G]] with mean $a \u003e 0$.\nThey are orthogonal for a weight on the nonnegative integers rather than on an interval.\nThey belong to the [[Askey scheme|RR2S05C09B]] as a limit of the [[Krawtchouk|CJGKW80KBR]] and [[Meixner polynomials|CEC3DESCFT]].\n\n## Explanations\n\n### L1 human\n\nCount random events, such as calls arriving at a desk in an hour, with an average of $a$ events.\nThe count follows the Poisson distribution, which puts weight $e^{-a} a^x / x!$ on each whole number $x$.\nThe Charlier polynomials are the polynomials in $x$ that are perpendicular to each other when one averages with this weight.\nThe first nonconstant one is $1 - x/a$, whose average is $1 - a/a = 0$.\n",
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