There are two analytic approaches to Bernoulli polynomials Bn (x): either by way of the generating function zexz/ (ez - 1) = ∑ Bn(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 zNexz/(ez - TN-1 (z)), where TN (z) denotes the N th Maclaurin polynomial of ez, 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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory