Polynomials m_k(x;β, c) which form the Sheffer sequence for g(t) | = | ((1 - c)/(1 - c e^t))^β f(t) | = | (1 - e^t)/(c^(-1) - e^t) and have generating function sum_(k = 0)^∞ (m_k(x;b, c))/(k!) t^k = (1 - t/c)^x (1 - t)^(-x - b). The are given in terms of the hypergeometric series by m_n(x;γ, μ) = (γ)_n _2 F_1(-n, - x;γ;1 - μ^(-1)), where (x)_n is the Pochhammer symbol.

Krawtchouk polynomial | Meixner polynomial of the second kind | Sheffer sequence

Josef Meixner