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Macdonald operators are well known as the 'commutative family' acting on thesymmetric 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 Dunkloperator appear especially as the coefficient of h^2. These Dunkl operatorsalso consist of commutative family. Then, as to the coefficient of h^3, it isnatural to expect that third-degree Dunkl operator appears. The object of thispaper is to calculate the coefficients of h^3 in the h-expansion of Macdonaldoperators explicitly, to introduce the method of calculation, and to prove thatthey can be expressed as the polynomials of Dunkl operators.

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