In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). Given a sequence, , the ordinary generating function (OGF) of the sequence, denoted , and the exponential generating function (EGF) of the sequence, denoted , are defined by the formal power series In this article, we use the convention that the ordinary (exponential) generating function for a sequence is denoted by the uppercase function / for some fixed or formal when the context of this notation is clear. Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by . The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and transformation formulas. Extracting arithmetic progressions of a sequence Series multisection provides formulas for generating functions enumerating the sequence given an ordinary generating function where , , and . In the first two cases where , we can expand these arithmetic progression generating functions directly in terms of : More generally, suppose that and that denotes the primitive root of unity. Then we have the following formula, often known as the root of unity filter: For integers , another useful formula providing somewhat reversed floored arithmetic progressions are generated by the identity Powers of an OGF and composition with functions The exponential Bell polynomials, , are defined by the exponential generating function The next formulas for powers, logarithms, and compositions of formal power series are expanded by these polynomials with variables in the coefficients of the original generating functions. The formula for the exponential of a generating function is given implicitly through the Bell polynomials by the EGF for these polynomials defined in the previous formula for some sequence of . Reciprocals of an OGF (special case of the powers formula) The power series for the reciprocal of a generating function, , is expanded by If we let denote the coefficients in the expansion of the reciprocal generating function, then we have