The Rademacher conjecture and <Emphasis Type="Italic">q</Emphasis>-partial fractions

In his classic book, Topics in Analytic Number Theory, H. Rademacher posed a natural conjecture concerning the generating function for p(n), the number of partitions of n. In this paper we undertake a systematic study of an expansion technique that has its genesis in the work of Cayley. We apply this to the Rademacher conjecture, and obtain the first positive result providing theoretical evidence for the conjecture.