A $q$-analogue of the type $A$ Dunkl operator and integral kernel

Details A $q$-analogue of the type $A$ Dunkl operator and integral kernel A $q$-analogue of the type $A$ Dunkl operator and integral kernel

1997-01-30

arxiv.org/abs/q-alg/9701039v1

Mathematics

Quantum Algebra

LaTeX 2.09, 15 pages

Scientific paper

We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of raising/lowering operators with a simple action on non-symmetric Macdonald polynomials. A bilinear series of non-symmetric Macdonald polynomials is introduced as a $q$-analogue of the type $A$ Dunkl integral kernel ${\cal K}_A(x;y)$. The aforementioned operators are used to show that the function satisfies $q$-analogues of the fundamental properties of ${\cal K}_A(x;y)$.

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