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\(q\)-Series with Applications to Combinatorics, Number Theory, and Physics
Edited by: Bruce C. Berndt, University of Illinois, Urbana, IL, and Ken Ono, University of Wisconsin, Madison, WI
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Contemporary Mathematics
2001; 277 pp; softcover
Volume: 291
ISBN-10: 0-8218-2746-4
ISBN-13: 978-0-8218-2746-8
List Price: US$80
Member Price: US$64
Order Code: CONM/291
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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

  • B. C. Berndt and K. Ono -- \(q\)-series Piano recital: Levis faculty center
  • Congruences and conjectures for the partition function
  • MacMahon's partition analysis VII: Constrained compositions
  • Crystal bases and \(q\)-identities
  • The Bailey-Rogers-Ramanujan group
  • Multiple polylogarithms: A brief survey
  • Swinnerton-Dyer type congruences for certain Eisenstein series
  • More generating functions for \(L\)-function values
  • On sums of an even number of squares, and an even number of triangular numbers: An elementary approah based on Ramanujan's \(_1\psi_1\) summation formula
  • Some remarks on multiple Sears transformations
  • Another way to count colored Frobenius partitions
  • Proof of a summation formula for an \(\tilde A_n\) basic hypergeometric series conjectured by Warnaar
  • On the representation of integers as sums of squares
  • 3-regular partitions and a modular K3 surface
  • A new look at Hecke's indefinite theta series
  • A proof of a multivariable elliptic summation formula conjectured by Warnaar
  • Multilateral transformations of \(q\)-series with quotients of parameters that are nonnegative integral powers of \(q\)
  • Completeness of basic trigonometric system in \(\mathcal{L}^{p}\)
  • The generalized Borwein conjecture. I. The Burge transform
  • Mock \(\vartheta\)-functions and real analytic modular forms
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