Integral operators and dual orthogonal systems on a half-line

A generalized Laplace transformation mu--> W-theta,W-mu, on the set of probability measures on R+, is introduced. The kernel of the transformation is chosen to be (w) over bar (theta)(zs) = F-0(1)(theta + 1; zs) (F-0(1) is the hypergeometric function, theta > -1, s is an element of R+, z is an element of C). A family of measures M-theta - {mu. : W-theta,W-mu is an element of L}, where L stands for the set of Laguerre entire functions is studied. The set L consists of polynomials with real nonpositive zeros only, as well as of their uniform limits on compact subsets of C. The set M-theta contains, among others, the Euler measure d(gammatheta) = (s(theta) /Gamma(theta + 1)) exp (-s) ds and the Dirac measures delta(s) x is an element of R+, which play a peculiar role in the Urbanik algebras defined by the transformation mu --> W-theta,W-mu. A sufficient condition for the measures dmu(s) = C(theta)s(theta) exp (- phi (s)) de to belong to M-theta, is given. For mu is an element of M-theta, integral operators with the kernels K-theta(mu) (z, s) = w(theta)(zs)/W-theta,W-mu (z), acting in the real Hilbert spaces L-2(R+, dmu), are studied. In particular, dual orthogonal systems of Appell polynomials are constructed.