Our purpose is to develop a systematic theory of formal power series. Such a theory is known, or at least presumed, by many writers on mathematics, who use it to avoid questions of convergence in infinite series. What is done here is to formulate the theory on a proper logical basis and thus to reveal the absence of the convergence question. Thus "hard" analysis can be replaced by "soft" analysis in many applications.
John Riordan [4] has discussed these matters in a chapter on generating functions, but his interest is in the applications to combinatorial problems. A more abstract discussion is given by de Branges and Rovnyak [l]. Many examples of the use of formal power series could be cited from the literature; we mention only two, one by John Riordan [5] the other by David Zeitlin [б].
The scheme of the paper is as follows. The theory of formal power series is developed in Sections 3, 4, 5, 6, 7, 11, and 12. Applications to number theory and combinatorial analysis are discussed in Sections 2, 8, 9, 10, and in the last part of 11.
The paper is self-contained insofar as it pertains to the theory of formal power series. However, in the applications of this theory, especially in the application to partitions in Section 9, we do not repeat here the fundamental results needed from number theory. Thus Sections 9 and 10 may be difficult for a reader who is not too familiar with the basic theory of partitions and the sum of divisors function. This difficulty can be removed by use of the specific references given in these sections; only a few pages of fairly straightforward material are needed as background. In Section 11 on the other hand, the background material is set forth in detail because the source is not too readily available.