Abstract
The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange-Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller-Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite-Miller-Lee polynomials.