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.

On the Proofs of the Rogers-Ramanujan Identities.- Bibasic Summation, Transformation and Expansion Formulas, q-Analogues of Clausen’s Formula, and Nonnegative Basic Hypergeometric Series.- Identities.- In the Land of Oz.- On the Gaussian Polynomials.- A One-line High School Algebra Proof of the Unimodality of the Gaussian Polynomials [nk] for k < 20.- An Elementary Proof of a q-Binomial Identity.- Some Macdonald-Mehta Integrals by Brute Force.- Macdonald Conjectures and the Selberg Integral.- Analogs and Extensions of Selberg’s Integral.- An Elementary Approach to the Macdonald Identities.- Continuous q-Hermite Polynomials when q > 1.- Generalized Rook Polynomials and Orthogonal Polynomials.- Monotonicity of Zeros of Orthogonal Polynomials.- Symmetry Techniques and Orthogonality for q-Series.