Macdonald Polynomials and Multivariable Basic Hypergeometric Series

Abstract

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 ${}_6phi_5$ summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised ${}_8phi_7$ 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.

Keywords

• Macdonald polynomials
• Pieri formula
• recursion formula
• matrix inversion
• basic hypergeometric series
• ${}_6phi_5$ summation
• Jackson's ${}_8phi_7$ summation
• $A_{n-1}$ series