A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
Pierpaolo Natalini. Angela Bernardini. "A generalization of the Bernoulli polynomials." J. Appl. Math. 2003 (3) 155 - 163, 5 March 2003. https://doi.org/10.1155/S1110757X03204101