A sequence of Appell polynomials is defined as a sequence \(A=\{a_n(x),n\geq 0\}\) of polynomials \(a_n(x)\) satisfying the properties: (a) have the degree equal to \(n\); (b) behave like power-law functions under differentiation, i.e. \({da_n(x) \over dx}= na_{-1}(x)\), \(n=0,1, \dots\). It is shown that the Appell polynomials satisfy a simple identity which forms the basis of many analytic results that appear, at first glance, to be disconnected. In particular, the author uniformly derives Taylor's expansion with remainder in Cauchy's form, the universal Bernoulli series, and the Euler-MacLaurin summation formula.