The theory of partitions, founded by Euler, has led in
a natural way to the idea of basic hypergeometric series, also known
as Eulerian series. These series were first studied systematically
by Heine, but many early results are attributed to Euler, Gauss,
and Jacobi. Today, research in $q$-hypergeometric series is
very active, and there are now major interactions with Lie
algebras, combinatorics, special functions, and number theory.
However, the theory has been developed to such an extent and
with such a profusion of powerful and general results that the subject
can appear quite formidable to the uninitiated. By providing a
simple approach to basic hypergeometric series, this book provides
an excellent elementary introduction to the subject.
The starting point is a simple function of several
variables satisfying a number of $q$-difference equations. The
author presents an elementary method for using these equations to
obtain transformations of the original function. A bilateral series,
formed from this function, is summed as an infinite product,
thereby providing an elegant and fruitful result which goes back to
Ramanujan. By exploiting a special case, the author is able to evaluate
the coefficients of several classes of infinite products in terms
of divisor sums. He also touches on general transformation theory
for basic series in many variables and the basic multinomial, which is
a generalization of a finite sum.
These developments lead naturally to the arithmetic domains
of partition theory, theorems of Liouville type, and sums of
squares. Contact is also made with the mock theta-functions of
Ramanujan, which are linked to the rank of partitions. The author gives
a number of examples of modular functions with multiplicative
coefficients, along with the beginnings of an elementary constructive
approach to the field of modular equations.
Requiring only an undergraduate background in mathematics, this book
provides a rapid entry into the field. Students of partitions, basic
series, theta-functions, and modular equations, as well as
research mathematicians interested in an elementary approach to these
areas, will find this book useful and enlightening. Because of the
simplicity of its approach and its accessibility, this work may prove
useful as a textbook.