  Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and
Bessel have the property that the polynomials of each system are eigenfunctions
of a second order ordinary differential operator. According to a classical
theorem by Bochner they are the only systems with this property. Similarly, the
polynomials of Charlier, Meixner, Kravchuk and Hahn are both orthogonal and are
eigenfunctions of a suitable difference operator of second order. We recall
that according to the famous theorem of Favard-Shohat, the condition of
orthogonality is equivalent to the 3-term recurrence relation. Vector
orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more
terms, according to a theorem by J. Van Iseghem and this characterizes them.
  Motivated by Bochner's theorem we are looking for VOP that are also
eigenfunctions of a differential (difference) operator. We call these
simultaneous conditions Bochner's property.
  The goal of this paper is to introduce methods for construction of VOP which
have Bochner's property. The methods are purely algebraic and are based on
automorphisms of non-commutative algebras. They also use ideas from the so
called bispectral problem. Applications of the abstract methods include broad
generalizations of the classical orthogonal polynomials, both continuous and
discrete. Other results connect different families of VOP, including the
classical ones, by linear transforms of purely algebraic origin, despite of the
fact that, when interpreted analytically, they are integral transformations.
