The h-expansion of Macdonald operators and their expression by Dunkl operators

Abstract

Macdonald operators are well known as the 'commutative family' acting on the symmetric functions over Q(q,t). If we suppose that q=exp(h) and t=exp(beta h) and observe the Taylor expansion around h=0, we can see the second-degree Dunkl operator appear especially as the coefficient of h^2. These Dunkl operators also consist of commutative family. Then, as to the coefficient of h^3, it is natural to expect that third-degree Dunkl operator appears. The object of this paper is to calculate the coefficients of h^3 in the h-expansion of Macdonald operators explicitly, to introduce the method of calculation, and to prove that they can be expressed as the polynomials of Dunkl operators.