  This paper sets the groundwork for the consideration of families of
recursively defined polynomials and rational functions capable of describing
the Bernoulli numbers. These families of functions arise from various recursive
definitions of the Bernoulli numbers. The derivation of these recursive
definitions is shown here using the original application of the Bernoulli
numbers: sums of powers of positive integers, i.e., Faulhaber's problem. Since
two of the three recursive forms of the Bernoulli numbers shown here are
readily available in literature and simple to derive, this paper focuses on the
development of the third, non-linear recursive definition arising from a
derivation of Faulhaber's formula. This definition is of particular interest as
it shows an example of an equivalence between linear and non-linear recursive
expressions. Part II of this paper will follow up with an in-depth look at this
non-linear definition and the conversion of the linear definitions into
polynomials and rational functions.
