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  "body": "\n## Definition\n\nLet $a \u003e 0$ be a real number and let $n \\ge 0$ be an integer.\nWrite $x^{\\underline{k}} = x (x - 1) \\cdots (x - k + 1)$ for the falling factorial, with $x^{\\underline{0}} = 1$.\nThe *Charlier polynomial* of degree $n$ with parameter $a$ is\n$$C_n(x; a) = \\sum_{k=0}^{n} \\binom{n}{k} \\left( -\\frac{1}{a} \\right)^k x^{\\underline{k}},$$\nwhere $\\binom{n}{k}$ is the [[binomial coefficient|0GBZM0MT0F]].\n\n## Remarks\n\nIn hypergeometric notation, $C_n(x; a) = {}_2F_0\\left( -n, -x; ; -\\frac{1}{a} \\right)$.\nThey are orthogonal for the [[Poisson distribution|0XB9GQTN9G]]: $\\sum_{x=0}^{\\infty} \\frac{a^x}{x!} C_m(x; a) C_n(x; a) = a^{-n} e^a n! \\, \\delta_{mn}$.\nExample: $C_1(x; a) = 1 - \\frac{x}{a}$.\nCheck: $\\sum_{x \\ge 0} \\frac{a^x}{x!} \\left( 1 - \\frac{x}{a} \\right) = e^a - \\frac{1}{a} \\cdot a e^a = 0$, and $\\sum_{x \\ge 0} \\frac{a^x}{x!} \\left( 1 - \\frac{x}{a} \\right)^2 = e^a \\left( 1 - 2 + \\frac{a + a^2}{a^2} \\right) = \\frac{e^a}{a}$.\nThey are symmetric in degree and variable: $C_n(x; a) = C_x(n; a)$ for nonnegative integers $n$ and $x$.\nNon-example: the monomials $1$ and $x$ are not orthogonal for this weight, since $\\sum_{x \\ge 0} \\frac{a^x}{x!} x = a e^a \\ne 0$.\n",
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