Charlier polynomials

Remarks

In hypergeometric notation, $C_n(x; a) = {}_2F_0\left( -n, -x; ; -\frac{1}{a} \right)$. They are orthogonal for the Poisson distribution: $\sum_{x=0}^{\infty} \frac{a^x}{x!} C_m(x; a) C_n(x; a) = a^{-n} e^a n! \, \delta_{mn}$. Example: $C_1(x; a) = 1 - \frac{x}{a}$. Check: $\sum_{x \ge 0} \frac{a^x}{x!} \left( 1 - \frac{x}{a} \right) = e^a - \frac{1}{a} \cdot a e^a = 0$, and $\sum_{x \ge 0} \frac{a^x}{x!} \left( 1 - \frac{x}{a} \right)^2 = e^a \left( 1 - 2 + \frac{a + a^2}{a^2} \right) = \frac{e^a}{a}$. They are symmetric in degree and variable: $C_n(x; a) = C_x(n; a)$ for nonnegative integers $n$ and $x$. Non-example: the monomials $1$ and $x$ are not orthogonal for this weight, since $\sum_{x \ge 0} \frac{a^x}{x!} x = a e^a \ne 0$.