Abstract
This chapter presents a study of the different kinds of series that are widely used as generating functions. It discusses formal power series. Operations on formal series involve corresponding operations on their coefficients. If the series actually converge and represent functions, operations on those functions correspond to certain operations on the power series coefficients of the expansions of those functions. The chapter explores a few of these relationships. It also discusses the calculus of formal exponential generating functions and reviews the basic analytic properties of power series and their coefficient sequences. Generating functionologists need reference lists of known power series and other series that occur frequently in applications of the theory. The chapter presents a list of some useful power series. For each series, it shows the series and its sum. It also discusses another kind of generating function that matches yet another kind of convolution of two sequences, a kind that also occurs naturally in many problems in combinatorics and number theory.
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