SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #10  -------------   OP-SF NET 9.1  -------------  January 15, 2002
From: OP-SF NET Editor 
Subject: q-Series Conference Proceedings


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


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.

Contents (authors names are not given)
     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 approach 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

                     Publisher: American Mathematical Society
                     Distributor: American Mathematical Society
                     Series: Contemporary Mathematics, ISSN: 0271-4132
                     Volume: 291
                     Publication Year: 2001
                     ISBN: 0-8218-2746-4
                     Paging: 277 pp.
                     Binding: Softcover
                     List Price: $69
                     Institutional Member Price: $55
                     Individual Member Price: $41
                     Order Code: CONM/291

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