Properties of four quintic theta functions are developed in parallel with
those of the classical Jacobi null theta functions. The quintic theta functions
are shown to satisfy analogues of Jacobi's quartic theta function identity and
counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of
Sign Formulas. The resulting library of quintic transformation formulas is used
to describe series multisections for modular forms in terms of simple matrix
operations. These efforts culminate in a formal technique for deducing
congruences modulo powers of five for a variety of combinatorial generating
functions, including the partition function. Further analysis of the quintic
theta functions is undertaken by exploring their modular properties and their
connection to Eisenstein series. The resulting relations lead to a coupled
system of differential equations for the quintic theta functions
