Identities and Approximation Formulas for Faulhaber'S Formula Revealing in Applications of Moment Generating Function, Distribution, and Arithmetic Functions

The aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special functions are given. By using the Riemann zeta function and the Liouville function, we derive a novel moment formula for probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, we derive formulas not only for the sum of powers of consecutive integers involving the Bernoulli polynomials, but also for the conditional probability involving the Laplace rule of succession.