Infinite order differential operators in spaces of entire functions. (Q1868776)

The authors deal with the differentiation with respect to a single complex variable \(z\in\mathbb{C}\) and the operators constructed by means of the following differential expression \[ \Delta_{\theta,\omega} = \Delta_\theta +\omega zD \overset {\text{def}} = (\theta+zD)D + \omega zD, \tag{1} \] where \(D=\frac {\partial} {\partial z}\) and \(\theta \geq 0\), \(\omega\in \mathbb{R}\) are parameters. Given entire functions \(\varphi, f:\mathbb{C}\to\mathbb{C}\) we set \[ \bigl(\varphi (\Delta_{\theta, \omega}) f \bigr) (z)= \sum^\infty_{k=1} \frac{1}{k!} \varphi^{(k)} (0)( \Delta^k_{\theta,\omega} f)(z). \tag{2} \] It is shown that, for \(\omega \geq 0\), such operators preserve the set of Laguerre entire functions provided the function \(\varphi\) also belongs to this set. Moreover an integral representation of \(\exp(a\Delta_{\theta,\omega})\), \(a>0\) is obtained.