Application of a composition of generating functions for obtaining explicit formulas of polynomials

Abstract: Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials.