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) = Sigma 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) - TN-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.