dilogarithm and trilogarithm functions, respectively), the alternating zeta function and the Riemann zeta function are formulated from the previous negative-order series results found in the references. In particular, when (or equivalently, when in the table above), we have the following special case series for the dilogarithm and corresponding constant value of the alternating zeta function: When (or when in the notation used in the previous subsection), we similarly obtain special case series for these functions given by It is known that the first-order harmonic numbers have a closed-form exponential generating function expanded in terms of the natural logarithm, the incomplete gamma function, and the exponential integral given by Additional series representations for the r-order harmonic number exponential generating functions for integers are formed as special cases of these negative-order derivative-based series transformation results. For example, the second-order harmonic numbers have a corresponding exponential generating function expanded by the series Generalized negative-order zeta series transformations A further generalization of the negative-order series transformations defined above is related to more Hurwitz-zeta-like, or Lerch-transcendent-like, generating functions. Specifically, if we define the even more general parametrized Stirling numbers of the second kind by , for non-zero such that , and some fixed , we have that Moreover, for any integers , we have the partial series approximations to the full infinite series in the previous equation given by Examples of the generalized negative-order zeta series transformations Series for special constants and zeta-related functions resulting from these generalized derivative-based series transformations typically involve the generalized r-order harmonic numbers defined by for integers . A pair of particular series expansions for the following constants when is fixed follow from special cases of BBP-type identities as Several other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include Additionally, we can give another new explicit series representation of the inverse tangent function through its relation to the Fibonacci numbers expanded as in the references by for and where the golden ratio (and its reciprocal) are respectively defined by . Inversion relations and generating function identities Inversion relations An inversion relation is a pair of equations of the form which is equivalent to the orthogonality relation Given two sequences, and , related by an inverse relation of the previous form, we sometimes seek to relate the OGFs and EGFs of the pair of sequences by functional equations implied by