  We define a new class of generating function transformations related to
polylogarithm functions, Dirichlet series, and Euler sums. These
transformations are given by an infinite sum over the $j^{th}$ derivatives of a
sequence generating function and sets of generalized coefficients satisfying a
non-triangular recurrence relation in two variables. The generalized
transformation coefficients share a number of analogous properties with the
Stirling numbers of the second kind and the known harmonic number expansions of
the unsigned Stirling numbers of the first kind.
  We prove a number of properties of the generalized coefficients which lead to
new recurrence relations and summation identities for the $k$-order harmonic
number sequences. Other applications of the generating function transformations
we define in the article include new series expansions for the polylogarithm
function, the alternating zeta function, and the Fourier series for the
periodic Bernoulli polynomials. We conclude the article with a discussion of
several specific new "almost" linear recurrence relations between the
integer-order harmonic numbers and the generalized transformation coefficients,
which provide new applications to studying the limiting behavior of the zeta
function constants, $\zeta(k)$, at integers $k \geq 2$.
