is expanded as a finite sum involving only paired reciprocals of the Laguerre polynomials in the form of Moreover, since the single factorial function is given by both and , we can generate the single factorial function terms using the approximate rational convergent generating functions up to order . This observation suggests an approach to approximating the exact (formal) Laplace–Borel transform usually given in terms of the integral representation from the previous section by a Hadamard product, or diagonal-coefficient, generating function. In particular, given any OGF we can form the approximate Laplace transform, which is -order accurate, by the diagonal coefficient extraction formula stated above given by Examples of sequences enumerated through these diagonal coefficient generating functions arising from the sequence factorial function multiplier provided by the rational convergent functions include where denotes a modified Bessel function, denotes the subfactorial function, denotes the alternating factorial function, and is a Legendre polynomial. Other examples of sequences enumerated through applications of these rational Hadamard product generating functions given in the article include the Barnes G-function, combinatorial sums involving the double factorial function, sums of powers sequences, and sequences of binomials. Derivative transformations Positive and negative-order zeta series transformations For fixed , we have that if the sequence OGF has derivatives of all required orders for , that the positive-order zeta series transformation is given by where denotes a Stirling number of the second kind. In particular, we have the following special case identity when when denotes the triangle of first-order Eulerian numbers: We can also expand the negative-order zeta series transformations by a similar procedure to the above expansions given in terms of the -order derivatives of some and an infinite, non-triangular set of generalized Stirling numbers in reverse, or generalized Stirling numbers of the second kind defined within this context. In particular, for integers , define these generalized classes of Stirling numbers of the second kind by the formula Then for and some prescribed OGF, , i.e., so that the higher-order derivatives of exist for all , we have that A table of the first few zeta series transformation coefficients, , appears below. These weighted-harmonic-number expansions are almost identical to the known formulas for the Stirling numbers of the first kind up to the leading sign on the weighted harmonic number terms in the expansions. Examples of the negative-order zeta series transformations The next series related to the polylogarithm functions (the