Ebook: q-Series and Partitions

This IMA Volume in Mathematics and its Applications q-Series and Partitions is based on the proceedings of a workshop which was an integral part of the 1987-88 IMA program on APPLIED COMBINATORICS. We are grateful to the Scientific Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for planning and implementing an exciting and stimulating year­ long program. We especially thank the Workshop Organizer, Dennis Stanton, for organizing a workshop which brought together many of the major figures in a variety of research fields in which q-series and partitions are used. A vner Friedman Willard Miller, Jr. PREFACE This volume contains the Proceedings of the Workshop on q-Series and Parti­ tions held at the IMA on March 7-11, 1988. Also included are papers by Goodman and O'Hara, Macdonald, and Zeilberger on unimodality. This work was of substan­ tial interest and discussed by many participants in the Workshop. The papers have been grouped into four parts: identities, unimodality of Gaus­ sian polynomials, constant term problems and related integrals, and orthogonal polynomials. They represent a cross section of the recent work on q-series includ­ ing: partitions, combinatorics, Lie algebras, analysis, and mathematical physics. I would like to thank the staff of the IMA, and its directors, Avner Friedman and Willard Miller, Jr., for providing a wonderful environment for the Workshop. Patricia Brick and Kaye Smith prepared the manuscripts.

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This volume contains the proceedings of the workshop held for the Applied Combinatorics program in March, 1988. The central idea of the workshop is the recent interplay of the classical analysis of q-series, and the combinatorial analysis of partitions of intergers. Many related topics are discussed, including orthogonal polynomials, the Macdonald conjectures for root systems, and related integrals. Those people interested in combinatorial enumeration and special functions will find this volume of interest. Recent applications of q-series (and related functions) to exactly solvable statistical mechanics models and to statistics makes this volume of interest to non-specialists. Included are several expository papers, and a series of papers on new work on the unimodality of the q-binomial coefficient.

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This volume contains the proceedings of the workshop held for the Applied Combinatorics program in March, 1988. The central idea of the workshop is the recent interplay of the classical analysis of q-series, and the combinatorial analysis of partitions of intergers. Many related topics are discussed, including orthogonal polynomials, the Macdonald conjectures for root systems, and related integrals. Those people interested in combinatorial enumeration and special functions will find this volume of interest. Recent applications of q-series (and related functions) to exactly solvable statistical mechanics models and to statistics makes this volume of interest to non-specialists. Included are several expository papers, and a series of papers on new work on the unimodality of the q-binomial coefficient.
Content:
Front Matter....Pages i-xi
On the Proofs of the Rogers-Ramanujan Identities....Pages 1-14
Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series....Pages 15-34
Identities....Pages 35-44
In the Land of OZ....Pages 45-55
On the Gaussian Polynomials....Pages 57-66
A One-Line High School Algebra Proof of the Unimodality of the Gaussian Polynomials [ k n ] for k < 20....Pages 67-72
An Elementary Proof of a q-Binomial Identity....Pages 73-75
Some Macdonald-Mehta Integrals by Brute Force....Pages 77-98
Macdonald Conjectures and the Selberg Integral....Pages 99-108
Analogs and Extensions of Selberg’s Integral....Pages 109-137
An Elementary Approach to the Macdonald Identities....Pages 139-149
Continuous q-Hermite Polynomials When q > 1....Pages 151-158
Generalized Rook Polynomials and Orthogonal Polynomials....Pages 159-176
Monotonicity of Zeros of Orthogonal Polynomials....Pages 177-190
Symmetry Techniques and Orthogonality for q-Series....Pages 191-212

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This volume contains the proceedings of the workshop held for the Applied Combinatorics program in March, 1988. The central idea of the workshop is the recent interplay of the classical analysis of q-series, and the combinatorial analysis of partitions of intergers. Many related topics are discussed, including orthogonal polynomials, the Macdonald conjectures for root systems, and related integrals. Those people interested in combinatorial enumeration and special functions will find this volume of interest. Recent applications of q-series (and related functions) to exactly solvable statistical mechanics models and to statistics makes this volume of interest to non-specialists. Included are several expository papers, and a series of papers on new work on the unimodality of the q-binomial coefficient.
Content:
Front Matter....Pages i-xi
On the Proofs of the Rogers-Ramanujan Identities....Pages 1-14
Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series....Pages 15-34
Identities....Pages 35-44
In the Land of OZ....Pages 45-55
On the Gaussian Polynomials....Pages 57-66
A One-Line High School Algebra Proof of the Unimodality of the Gaussian Polynomials [ k n ] for k < 20....Pages 67-72
An Elementary Proof of a q-Binomial Identity....Pages 73-75
Some Macdonald-Mehta Integrals by Brute Force....Pages 77-98
Macdonald Conjectures and the Selberg Integral....Pages 99-108
Analogs and Extensions of Selberg’s Integral....Pages 109-137
An Elementary Approach to the Macdonald Identities....Pages 139-149
Continuous q-Hermite Polynomials When q > 1....Pages 151-158
Generalized Rook Polynomials and Orthogonal Polynomials....Pages 159-176
Monotonicity of Zeros of Orthogonal Polynomials....Pages 177-190
Symmetry Techniques and Orthogonality for q-Series....Pages 191-212

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This volume contains the proceedings of the workshop held for the Applied Combinatorics program in March, 1988. The central idea of the workshop is the recent interplay of the classical analysis of q-series, and the combinatorial analysis of partitions of intergers. Many related topics are discussed, including orthogonal polynomials, the Macdonald conjectures for root systems, and related integrals. Those people interested in combinatorial enumeration and special functions will find this volume of interest. Recent applications of q-series (and related functions) to exactly solvable statistical mechanics models and to statistics makes this volume of interest to non-specialists. Included are several expository papers, and a series of papers on new work on the unimodality of the q-binomial coefficient.
Content:
Front Matter....Pages i-xi
On the Proofs of the Rogers-Ramanujan Identities....Pages 1-14
Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series....Pages 15-34
Identities....Pages 35-44
In the Land of OZ....Pages 45-55
On the Gaussian Polynomials....Pages 57-66
A One-Line High School Algebra Proof of the Unimodality of the Gaussian Polynomials [ k n ] for k < 20....Pages 67-72
An Elementary Proof of a q-Binomial Identity....Pages 73-75
Some Macdonald-Mehta Integrals by Brute Force....Pages 77-98
Macdonald Conjectures and the Selberg Integral....Pages 99-108
Analogs and Extensions of Selberg’s Integral....Pages 109-137
An Elementary Approach to the Macdonald Identities....Pages 139-149
Continuous q-Hermite Polynomials When q > 1....Pages 151-158
Generalized Rook Polynomials and Orthogonal Polynomials....Pages 159-176
Monotonicity of Zeros of Orthogonal Polynomials....Pages 177-190
Symmetry Techniques and Orthogonality for q-Series....Pages 191-212
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