Poisson-Charlier Polynomial

Polynomials $p_n(x)$ which belong to the distribution $d\alpha(x)$ where $\alpha(x)$ is a Step Function with Jump

$$j(x)=e^{-a}a^x(x!)^{-1}$$ (1)

at $x=0$, 1, ...for $a>0$.

$$p_n(x) = a^{n/2}(n!)^{-1/2} \sum_{\nu=0}^n (-1)^{n-\nu} {n\choose\nu} \nu! a^{-\nu}{x\choose\nu}$$ (2)

$$= a^{n/2}(n!)^{-1/2}(-1)^n[j(x)]^{-1}\Delta^n j(x-n)$$ (3)

$$= a^{-n/2}\sqrt{n!}\,L_n^{x-n}(a),$$ (4)

where ${n\choose k}$ is a Binomial Coefficient, $L_n^k(x)$ is an associated Laguerre Polynomial, and

$$\Delta f(x) = f(x+1)-f(x)$$ (5)

$$\Delta^n f(x) = \Delta[\Delta^{n-1} f(x)]= f(x+n)-{n\choose 1}f(x+n-1)+\ldots+(-1)^n f(x).$$ (6)

See also Poisson-Charlier Function

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 34-35, 1975.