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

Edited by: Bruce C. Berndt University of Illinois, Urbana, IL
Ken Ono University of Wisconsin, Madison, WI
Available Formats:
Electronic ISBN: 978-0-8218-7881-1
Product Code: CONM/291.E
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Click above image for expanded view$q$-Series with Applications to Combinatorics, Number Theory, and Physics Edited by: Bruce C. Berndt University of Illinois, Urbana, IL Ken Ono University of Wisconsin, Madison, WI Available Formats:  Electronic ISBN: 978-0-8218-7881-1 Product Code: CONM/291.E  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
• Book Details

Contemporary Mathematics
Volume: 2912001; 277 pp
MSC: Primary 05; 11; 33; 81; 82;

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.

Graduate students and research mathematicians interested in number theory.

• Articles
• Scott Ahlgren and Ken Ono - Congruences and conjectures for the partition function [ MR 1874518 ]
• George E. Andrews, Peter Paule and Axel Riese - MacMahon’s partition analysis. VII. Constrained compositions [ MR 1874519 ]
• Masato Okado, Anne Schilling and Mark Shimozono - Crystal bases and $q$-identities [ MR 1874520 ]
• D. Stanton - The Bailey-Rogers-Ramanujan group [ MR 1874521 ]
• Douglas Bowman and David M. Bradley - Multiple polylogarithms: a brief survey [ MR 1874522 ]
• Matthew Boylan - Swinnerton-Dyer type congruences for certain Eisenstein series [ MR 1874523 ]
• Gwynneth G. H. Coogan - More generating functions for $L$-function values [ MR 1874524 ]
• Shaun Cooper - 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 [ MR 1874525 ]
• Yasushi Kajihara - Some remarks on multiple Sears transformations [ MR 1874526 ]
• Louis W. Kolitsch - Another way to count colored Frobenius partitions [ MR 1874527 ]
• C. Krattenthaler - Proof of a summation formula for an $\tilde {A}_n$ basic hypergeometric series conjectured by Warnaar [ MR 1874528 ]
• Zhi-Guo Liu - On the representation of integers as sums of squares [ MR 1874529 ]
• Jeremy Lovejoy and David Penniston - $3$-regular partitions and a modular $K3$ surface [ MR 1874530 ]
• A. Polishchuk - A new look at Hecke’s indefinite theta series [ MR 1874531 ]
• Hjalmar Rosengren - A proof of a multivariable elliptic summation formula conjectured by Warnaar [ MR 1874532 ]
• Michael Schlosser - Multilateral transformations of $q$-series with quotients of parameters that are nonnegative integral powers of $q$ [ MR 1874533 ]
• Sergei K. Suslov - Completeness of basic trigonometric system in $\scr L^p$ [ MR 1874534 ]
• S. Ole Warnaar - The generalized Borwein conjecture. I. The Burge transform [ MR 1874535 ]
• S. P. Zwegers - Mock $\theta$-functions and real analytic modular forms [ MR 1874536 ]
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2912001; 277 pp
MSC: Primary 05; 11; 33; 81; 82;

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.

Graduate students and research mathematicians interested in number theory.

• Articles
• Scott Ahlgren and Ken Ono - Congruences and conjectures for the partition function [ MR 1874518 ]
• George E. Andrews, Peter Paule and Axel Riese - MacMahon’s partition analysis. VII. Constrained compositions [ MR 1874519 ]
• Masato Okado, Anne Schilling and Mark Shimozono - Crystal bases and $q$-identities [ MR 1874520 ]
• D. Stanton - The Bailey-Rogers-Ramanujan group [ MR 1874521 ]
• Douglas Bowman and David M. Bradley - Multiple polylogarithms: a brief survey [ MR 1874522 ]
• Matthew Boylan - Swinnerton-Dyer type congruences for certain Eisenstein series [ MR 1874523 ]
• Gwynneth G. H. Coogan - More generating functions for $L$-function values [ MR 1874524 ]
• Shaun Cooper - 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 [ MR 1874525 ]
• Yasushi Kajihara - Some remarks on multiple Sears transformations [ MR 1874526 ]
• Louis W. Kolitsch - Another way to count colored Frobenius partitions [ MR 1874527 ]
• C. Krattenthaler - Proof of a summation formula for an $\tilde {A}_n$ basic hypergeometric series conjectured by Warnaar [ MR 1874528 ]
• Zhi-Guo Liu - On the representation of integers as sums of squares [ MR 1874529 ]
• Jeremy Lovejoy and David Penniston - $3$-regular partitions and a modular $K3$ surface [ MR 1874530 ]
• A. Polishchuk - A new look at Hecke’s indefinite theta series [ MR 1874531 ]
• Hjalmar Rosengren - A proof of a multivariable elliptic summation formula conjectured by Warnaar [ MR 1874532 ]
• Michael Schlosser - Multilateral transformations of $q$-series with quotients of parameters that are nonnegative integral powers of $q$ [ MR 1874533 ]
• Sergei K. Suslov - Completeness of basic trigonometric system in $\scr L^p$ [ MR 1874534 ]
• S. Ole Warnaar - The generalized Borwein conjecture. I. The Burge transform [ MR 1874535 ]
• S. P. Zwegers - Mock $\theta$-functions and real analytic modular forms [ MR 1874536 ]
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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