RADIAL HEAT OPERATORS ON JACOBI-LIKE FORMS

We consider a differential operator D^X _λ associated to an integer λ acting on the space of formal power series, which may be regarded as the heat operator with respect to the radial coordinate in the 2λ-dimensional space for λ > 0. We show that D^X _λ carries Jacobilike forms of weight λ to ones of weight λ+2 and obtain the formula for the m-fold composite (D^X _λ )^[m] of such operators. We then determine the corresponding operators on modular series and as well as on automorphic pseudodifferential operators.