Heat Operators

Abstract

In this chapter we consider a differential operator \( \mathcal{D}_{\mu }^{X} \) associated to a real number μ acting on the space of formal power series, which may be regarded as the heat operator with respect to the radial coordinate in the 2μ-dimensional space for a positive integer μ. If λ is an integer, we show that \( \mathcal{D}_{\mu }^{X} \) carries Jacobi-like forms of weight λ to ones of weight λ + 2 and obtain the formula for the m-fold composite \( (\mathcal{D}_{\mu }^{X} )^{[m]} \) of such operators. We then determine the corresponding operators on modular series and as well as on automorphic pseudodifferential operators (cf. [72]).