transformation integral for the Stirling numbers of the second kind given as an example above. In particular, since we can use a variant of the positive-order derivative-based OGF transformations defined in the next sections involving the Stirling numbers of the second kind to obtain an integral formula for the generating function of the sequence, , and then perform a sum over the derivatives of the formal OGF, to obtain the result in the previous equation where the arithmetic progression generating function at hand is denoted by for each fixed . Hadamard products and diagonal generating functions We have an integral representation for the Hadamard product of two generating functions, and , stated in the following form: where I is the imaginary unit. More information about Hadamard products as diagonal generating functions of multivariate sequences and/or generating functions and the classes of generating functions these diagonal OGFs belong to is found in Stanley's book. The reference also provides nested coefficient extraction formulas of the form which are particularly useful in the cases where the component sequence generating functions, , can be expanded in a Laurent series, or fractional series, in , such as in the special case where all of the component generating functions are rational, which leads to an algebraic form of the corresponding diagonal generating function. Example: Hadamard products of rational generating functions In general, the Hadamard product of two rational generating functions is itself rational. This is seen by noticing that the coefficients of a rational generating function form quasi-polynomial terms of the form where the reciprocal roots, , are fixed scalars and where is a polynomial in for all . For example, the Hadamard product of the two generating functions and is given by the rational generating function formula Example: Factorial (approximate Laplace) transformations Ordinary generating functions for generalized factorial functions formed as special cases of the generalized rising factorial product functions, or Pochhammer k-symbol, defined by where is fixed, , and denotes the Pochhammer symbol are generated (at least formally) by the Jacobi-type J-fractions (or special forms of continued fractions) established in the reference. If we let denote the convergent to these infinite continued fractions where the component convergent functions are defined for all integers by and where denotes an associated Laguerre polynomial, then we have that the convergent function, , exactly enumerates the product sequences, , for all . For each , the convergent function