Summary: A generalized Laplace transformation \(\mu\mapsto W_{\theta,\mu}\), on the set of probability measures on \(\mathbb{R}_+\), is introduced. The kernel of the transformation is chosen to be \(\widetilde w_\theta (zs)={_0 F_1} (\theta+1; zs)\) \((_0F_1\) is the hypergeometric function, \(\theta>-1\), \(s\in \mathbb{R}_+\), \(x\in\mathbb{C})\). A family of measures \({\mathcal M}_\theta= \{\mu:W_{\theta, \mu}\in {\mathcal L}\}\), where \({\mathcal L}\) stands for the set of Laguerre entire functions, is studied. The set \({\mathcal L}\) consists of polynomials with real nonpositive zeros only, as well as of their uniform limits on compact subsets of \(\mathbb{C}\). The set \({\mathcal M}_\theta\) contains, among others, the Euler measure \(d \gamma_\theta= (s^\theta/ \Gamma(\theta+1)) \exp(-s)ds\) and the Dirac measures \(\delta_x\), \(x\in\mathbb{R}_+\) which play a peculiar role in the Urbanik algebras defined by the transformation \(\mu\mapsto W_{\theta,\mu}\).
A sufficient condition for the measures \(d\mu(s)= C_\theta s^\theta \exp(-\Phi(s)) ds\) to belong to \({\mathcal M}_\theta\), is given. For \(\mu\in{\mathcal M}_\theta\), integral operators with the kernels \(K^\mu_\theta (z,s)= w_\theta(zs)/W_{\theta, \mu}(z)\), acting in the real Hilbert spaces \(L^2(\mathbb{R}_+, d\mu)\), are studied. In particular, dual orthogonal systems of Appell polynomials are constructed.