Special functions and characterizations of probability distributions by zero regression properties

Characterizations of the binomial, negative binomial, gamma, Poisson, and normal distributions are obtained by the property of zero regression of certain polynomial statistics of arbitrary degree, on the mean. In each case, the equations which express zero regression are derived from the recurrence relations of a set of special functions. The differential recurrence relations of these special functions are used in the proofs of the characterization theorems.