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)z n /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 N e xz /(e z - T N-1 (z)), where T N (z) denotes the Nth 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.