# \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics

Share this page *Edited by *
*Bruce C. Berndt; Ken Ono*

The subject of \(q\)-series can be said to begin with Euler
and his
pentagonal number theorem. In fact, \(q\)-series are sometimes called
Eulerian
series. Contributions were made by Gauss, Jacobi, and Cauchy, but the first
attempt at a systematic development, especially from the point of view of
studying series with the products in the summands, was made by E. Heine in
1847. In the latter part of the nineteenth and in the early part of the
twentieth centuries, two English mathematicians, L. J. Rogers and F. H.
Jackson, made fundamental contributions.

In 1940, G. H. Hardy described what we now call Ramanujan's famous
\(_1\psi_1\) summation theorem as “a remarkable formula with many
parameters.”
This is now one of the fundamental theorems of the subject.

Despite humble beginnings, the subject of \(q\)-series has
flourished in the
past three decades, particularly with its applications to combinatorics,
number
theory, and physics. During the year 2000, the University of Illinois
embraced The Millennial Year in Number Theory. One of the events
that year was
the conference \(q\)-Series with Applications to Combinatorics,
Number Theory,
and Physics. This event gathered mathematicians from the world over to
lecture
and discuss their research.

This volume presents nineteen of the papers presented at the conference.
The
excellent lectures that are included chart pathways into the future and
survey the numerous applications of \(q\)-series to combinatorics,
number theory, and physics.

#### Readership

Graduate students and research mathematicians interested in number theory.

# Table of Contents

## $q$-Series with Applications to Combinatorics, Number Theory, and Physics

- Contents v6 free
- Preface vii8 free
- Program for q-series piano recital ix10 free
- Congruences and conjectures for the partition function 112 free
- MacMahon's partition analysis VII: Constrained compositions 1122
- Crystal bases and q-identities 2940
- The Bailey-Rogers-Ramanujan group 5566
- Multiple polylogarithms: A brief survey 7182
- Swinnerton-Dyer type congruences for certain Eisenstein series 93104
- More generating functions for L-function values 109120
- On sums of an even number of squares, and an even number of triangular numbers: an elementary approach based on Ramanujan's 1ψ1 summation formula 115126
- Some remarks on multiple Sears transformations 139150
- Another way to count colored Frobenius partitions 147158
- Proof of a summation formula for an Ãn basic hypergeometric series conjectured by Warnaar 153164
- On the representation of integers as sums of squares 163174
- 3-regular partitions and a modular K3 surface 177188
- A new look at Heeke's indefinite theta series 183194
- A proof of a multivariable elliptic summation formula conjectured by Warnaar 193204
- Multilateral transformations of q-series with quotients of parameters that are nonnegative integral powers of q 203214
- Completeness of basic trigonometric system in Lp 229240
- The generalized Borwein conjecture. I. The Burge transform 243254
- Mock v-functions and real analytic modular forms 269280