The Computation of Expected Values and Moments of Special Polynomials via Characteristic and Generating Functions

摘要

We give a brief summary and survey of the theory of characteristic function, moment generating functions, and probability generating functions. By using these functions and Sun's expectation operator for the Bernoulli polynomials, the Euler polynomials and the Hermite polynomials, we give some relations and identities, which are related to the expected values and moments of the random variables of the Laplace distribution, the Bernoulli polynomials, the Euler polynomials and the Hermite polynomials. Moreover, by using characteristics functions and generating functions, we also study the expected values of the Laplace distribution, the normal distribution, and the Genocchi numbers.