Hypergeometric Bernoulli polynomials and appell sequences

There are two analytic approaches to Bernoulli polynomials B_n (x): either by way of the generating function ze^xz/ (e^z - 1) = ∑ B_n(x)zⁿ/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function z^Ne^xz/(e^z - T_N-1 (z)), where T_N (z) denotes the N th Maclaurin polynomial of e^z, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.