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

Abstract

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 ${\cal K}_A(x;y)$.