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This paper sets the groundwork for the consideration of families ofrecursively defined polynomials and rational functions capable of describingthe Bernoulli numbers. These families of functions arise from various recursivedefinitions of the Bernoulli numbers. The derivation of these recursivedefinitions is shown here using the original application of the Bernoullinumbers: sums of powers of positive integers, i.e., Faulhaber's problem. Sincetwo of the three recursive forms of the Bernoulli numbers shown here arereadily available in literature and simple to derive, this paper focuses on thedevelopment of the third, non-linear recursive definition arising from aderivation of Faulhaber's formula. This definition is of particular interest asit shows an example of an equivalence between linear and non-linear recursiveexpressions. Part II of this paper will follow up with an in-depth look at thisnon-linear definition and the conversion of the linear definitions intopolynomials and rational functions.

Faulhaber's Problem Revisited: Alternate Methods for Deriving the Bernoulli Numbers--Part I