Spectral analysis of generators of representations of the group $U(3)$

A basis of eigen-vectors of the generator producing the Wigner $d$-functions is constructed in the space of irreducible unitary representation of the $U(3)$-group. Eigenvectors of the generator are described in terms of a certain class of orthogonal polynomials in two discrete variables which are generalizations of the Kravchuk polynomials. Continuous analogues of these polynomials are studied. Decomposition of the $d$-functions into exponential functions is obtained.