q-Series and Partitions

1. Front Matter

   Pages i-xi

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2. On the Proofs of the Rogers-Ramanujan Identities

   • George E. Andrews

   Pages 1-14

3. Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series

   • George Gasper

   Pages 15-34

4. Identities

   • Doron Zeilberger

   Pages 35-44

5. In the Land of OZ

   • David M. Bressoud

   Pages 45-55

6. On the Gaussian Polynomials

   • Frederick M. Goodman, Kathleen M. O’Hara

   Pages 57-66

7. A One-Line High School Algebra Proof of the Unimodality of the Gaussian Polynomials [ k n ] for k < 20

   • Doron Zeilberger

   Pages 67-72

8. An Elementary Proof of a q-Binomial Identity

   • I. G. Macdonald

   Pages 73-75

9. Some Macdonald-Mehta Integrals by Brute Force

   • Frank G. Garvan

   Pages 77-98

10. Macdonald Conjectures and the Selberg Integral

    • Laurent Habsieger

    Pages 99-108

11. Analogs and Extensions of Selberg’s Integral

    • Donald St. P. Richards

    Pages 109-137

12. An Elementary Approach to the Macdonald Identities

    • Dennis Stanton

    Pages 139-149

13. Continuous q-Hermite Polynomials When q > 1

    • Richard Askey

    Pages 151-158

14. Generalized Rook Polynomials and Orthogonal Polynomials

    • Ira M. Gessel

    Pages 159-176

15. Monotonicity of Zeros of Orthogonal Polynomials

    • Mourad E. H. Ismail

    Pages 177-190

16. Symmetry Techniques and Orthogonality for q-Series

    • Willard Miller Jr.

    Pages 191-212

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.

• Combinatorics
• Partition
• differential equation
• function
• orthogonal polynomials

• School of Mathematics, University of Minnesota, Minneapolis, USA

  Dennis Stanton

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