Compositio Mathematica

(Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula

a1 Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637-1546, U.S.A. e-mail:

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okounkov a   [Google Scholar] 


We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.

Key Words: Macdonald polynomials; shifted Macdonald polynomials; combinatorial formula; q-integral representation..