Macdonald Polynomials and Multivariable Basic Hypergeometric Series
Copyright policy )Macdonald Polynomials and Multivariable Basic Hypergeometric Series
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 6-phi-5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8-phi-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.
This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
1010 Mathematics, \(A_{n-1}\) series, 1010 Mathematik, $A_{n-1}$ series, FOS: Physical sciences, \({}_6\phi_5\) summation, matrix inversion, 33D52 (Primary) 15A09, 33D67 (Secondary), Pieri formula, Mathematics - Quantum Algebra, QA1-939, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, Quantum Algebra (math.QA), Jackson's ${}_8phi_7$ summation, Mathematical Physics, Basic hypergeometric functions associated with root systems, basic hypergeometric series, Mathematical Physics (math-ph), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), recursion formula, Jackson's \({}_6\phi_5\) summation, Mathematics - Classical Analysis and ODEs, Macdonald polynomials, Combinatorics (math.CO), ${}_6phi_5$ summation, Mathematics
1010 Mathematics, \(A_{n-1}\) series, 1010 Mathematik, $A_{n-1}$ series, FOS: Physical sciences, \({}_6\phi_5\) summation, matrix inversion, 33D52 (Primary) 15A09, 33D67 (Secondary), Pieri formula, Mathematics - Quantum Algebra, QA1-939, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, Quantum Algebra (math.QA), Jackson's ${}_8phi_7$ summation, Mathematical Physics, Basic hypergeometric functions associated with root systems, basic hypergeometric series, Mathematical Physics (math-ph), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), recursion formula, Jackson's \({}_6\phi_5\) summation, Mathematics - Classical Analysis and ODEs, Macdonald polynomials, Combinatorics (math.CO), ${}_6phi_5$ summation, Mathematics
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