The Poisson-Charlier polynomials c_k(x;a) form a Sheffer sequence with g(t) | = | e^(a(e^t - 1)) f(t) | = | a(e^t - 1), giving the generating function sum_(k = 0)^∞ (c_k(x;a))/(k!) t^k = e^(-t) ((a + t)/a)^x. The Sheffer identity is c_n(x + y;a) = sum_(k = 0)^n(n k) a^(k - n) c_k(y;a)(x)_(n - k), where (x)_n is a falling factorial. The polynomials satisfy the recurrence relation c_(n + 1)(x;a) = a^(-1) x c_n(x - 1;a) - c_n(x;a).

Laguerre polynomial | Poisson-Charlier function | Sheffer sequence