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Radial heat operators on Jacobi-like forms. (English) Zbl 1257.11047
Jacobi-like forms are formal power series, whose coefficients are holomorphic functions on the Poincaré upper half plane, satisfying a certain transformation formula with respect to the action of a discrete subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb R)\). The coefficients of a Jacobi-like form are related to modular forms for \(\Gamma\).
For a natural number \(\lambda\), the author considers a differential operator \(D_\lambda^X\) on the space of formal power series having the form of the radial heat operator on the \(2\lambda\)-dimensional space. It is shown that \(D_\lambda^X\) transforms Jacobi-like forms of weight \(\lambda\) into ones of weight \(\lambda +2\). An explicit formula for the composition \[ (D_\lambda^X)^{[m]}=D_{\lambda +2m-2}^X\circ \cdots \circ D_{\lambda +2}^X\circ D_\lambda^X \] is found and used to study its action on modular forms and automorphic pseudo-differential operators.

11F50 Jacobi forms
11F11 Holomorphic modular forms of integral weight
35S99 Pseudodifferential operators and other generalizations of partial differential operators