Bernoulli numbers and symmetric functions

Abstract

Some of the known properties of the Bernoulli numbers can be derived as specializations of the fundamental relationships between complete and elementary symmetric functions. In this paper, we introduce an infinite family of relationships between complete and elementary symmetric functions. As specializations of this result we derive connections between some lacunary recurrence relations with gaps of length 2r for the Bernoulli numbers and the integer partitions into at most $$r-1$$ parts.