we can use the previous integral representation together with the formula for extracting arithmetic progressions from a sequence OGF given above, to formulate the next integral representation for the so-termed modified Stirling number EGF as which is convergent provided suitable conditions on the parameter . Example: An EGF formula for the higher-order derivatives of the geometric series For fixed non-zero defined such that , let the geometric series over the non-negative integral powers of be denoted by . The corresponding higher-order derivatives of the geometric series with respect to are denoted by the sequence of functions for non-negative integers . These derivatives of the ordinary geometric series can be shown, for example by induction, to satisfy an explicit closed-form formula given by for any whenever . As an example of the third OGF EGF conversion formula cited above, we can compute the following corresponding exponential forms of the generating functions : Fractional integrals and derivatives Fractional integrals and fractional derivatives (see the main article) form another generalized class of integration and differentiation operations that can be applied to the OGF of a sequence to form the corresponding OGF of a transformed sequence. For we define the fractional integral operator (of order ) by the integral transformation which corresponds to the (formal) power series given by For fixed defined such that , we have that the operators . Moreover, for fixed and integers satisfying we can define the notion of the fractional derivative satisfying the properties that and for where we have the semigroup property that only when none of is integer-valued. Polylogarithm series transformations For fixed , we have that (compare to the special case of the integral formula for the Nielsen generalized polylogarithm function defined in) Notice that if we set , the integral with respect to the generating function, , in the last equation when corresponds to the Dirichlet generating function, or DGF, , of the sequence of provided that the integral converges. This class of polylogarithm-related integral transformations is related to the derivative-based zeta series transformations defined in the next sections. Square series generating function transformations For fixed non-zero such that and , we have the following integral representations for the so-termed square series generating function associated with the sequence , which can be integrated termwise with respect to : This result, which is proved in the reference, follows from a variant of the double factorial function