appropriately appreciated, and now the subject is a fundamental and active branch
of analysis. Many theorems in the theory of hypergeometric series have analogues
in basic or q-hypergeometric series, and undoubtedly Chaundy was referring to
what he considered to be the esoteric pastime of finding such q-analogues. But
irrespective of whether q-analogues exist or not, the subject is replete with beautiful,
elegant, and sometimes surprising theorems. Secondly, beginning with the work of
Jacobi, q-series have found applications to number theory, and the closeness of
these two subjects continues to be ever stronger. Thirdly, increasingly numerous
applications to combinatorics are being made, especially in the theory of partitions.
Fourthly, through the pioneering work of Rodney Baxter, Barry McCoy, and other
theoretical physicists in the past three decades, q-series are now a necessary tool in
their subject.
During the year 2000, the Mathematics Department at the University of Illi-
nois embraced The Millennial Year in Number Theory. In view of the increasing
importance and visibility of q-series, it seemed appropriate that as one of the events
in this auspicious Year, a conference, q-series with Applications to Combinatorics,
Number Theory, and Physics, should be held. On October 26-28, sixty-two math-
ematicians representing twelve countries gathered at the University of Illinois to
lecture about and discuss the latest findings. It also seemed appropriate to em-
phasize survey lectures to help us better chart a course for the future. A total of
thirty-nine lectures, highlighted by five plenary survey talks, were given. The ple-
nary lecturers were Scott Ahlgren, George Andrews, Richard Askey, Anne Schilling,
and Dennis Stanton. All four aspects (analysis, combinatorics, number theory, and
physics) of the subject were featured in these five lectures, as well as in the shorter
talks. All the participants helped to make the conference a very successful one, and
we thank all of them for their participation. These proceedings contain nineteen of
the papers presented at the conference. We hope that they will convey to readers
the richness, beauty, and efficacy of the subject.
Special thanks are given to Christian Krattenthaler for a superb evening piano
recital. His program can be found before the first paper in this volume.
Many organizations graciously helped to finance the conference. In particular,
we are grateful to the National Science Foundation, the National Security Agency,
the Number Theory Foundation, The University of Illinois, the Alfred P. Sloan
Foundation, and the David and Lucile Packard Foundation for their generous sup-
Bruce C. Berndt
Ken Ono
July, 2001
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