Zeta series generating function transformations related to generalized Stirling numbers and partial sums of the Hurwitz zeta function

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences as the partial sums of the Hurwitz zeta function. These transformation coefficients satisfy many properties which are analogous to known identities and expansions of the Stirling numbers of the first kind and to the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers s ≥ 3.