An explicit formula for generalized potential polynomials and its applications

Abstract

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol–Bernoulli, Apostol–Euler polynomials and the polynomials A n ( z ) ( τ ) involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers A n ( z ) , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.