Polynomials M_k(x) which form the associated Sheffer sequence for f(t) = (e^t - 1)/(e^t + 1) and have the generating function sum_(k = 0)^∞ (M_k(x))/(k!) t^k = ((1 + t)/(1 - t))^x. An explicit formula is given by M_n(x) = sum_(k = 0)^n(n k)(n - 1)_(n - k) 2^k (x)_k, where (x)_n is a falling factorial, which can be summed in closed form in terms of the hypergeometric function, gamma function, and polygamma function.

Pidduck polynomial

Magnus Gösta Mittag-Leffler