Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 056, 30 pages      math.CO/0611639
Contribution to the Vadim Kuznetsov Memorial Issue

Macdonald Polynomials and Multivariable Basic Hypergeometric Series

Michael J. Schlosser
Fakultät für Mathematik, Universität Wien, Nordbergstraàe 15, A-1090 Vienna, Austria

Received November 21, 2006; Published online March 30, 2007

We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6φ5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8φ7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexed by partitions with complex parts. These appear to possess nice properties.

Key words: Macdonald polynomials; Pieri formula; recursion formula; matrix inversion; basic hypergeometric series; 6φ5 summation; Jackson's 8φ7 summation; An-1 series.

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