To the theory of the $C_0$-operator orthogonal polynomials

Operator orthogonal polynomials are considered whose argument is the generator of a strongly continuous semigroup of transformations of class $C_0$ acting in a Banach space. Earlier such polynomials were considered by the authors in the case of the Chebyshev polynomials of the first and second kind. In this article more general classes of operator orthogonal polynomials are considered, which include the Jacobi and Aptekarev polynomials. Integral representation of operator fractional-rational functions and also Bessel operator functions of imaginary argument are presented.