  New sequences of orthogonal polynomials with ultra-exponential weight
functions are discovered. In particular, it gives an explicit solution to the
Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit
representations, generating functions and Rodrigues type formulae are derived.
The method is based on differential properties of the involved special
functions and their representations in terms of the Mellin-Barnes and Laplace
integrals. A notion of the composition polynomial orthogonality is introduced.
The corresponding advantages of this orthogonality to discover new sequences of
polynomials and their relations to the corresponding multiple orthogonal
polynomial ensembles are shown.
