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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.

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