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

TL;DR

This paper introduces a $q$-analogue of type $A$ Dunkl operators and constructs a corresponding integral kernel, extending the classical theory to the quantum setting and enabling new algebraic and analytical tools.

Contribution

It develops a $q$-analogue of type $A$ Dunkl operators and constructs a $q$-analogue of the Dunkl integral kernel using non-symmetric Macdonald polynomials.

Findings

Defined $q$-analogue Dunkl operators for type $A$

Constructed a $q$-analogue of the Dunkl integral kernel

Established properties of the $q$-kernel analogous to classical case

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