Orthonormal bases of polynomials in one complex variable

Let a sequence (P_n) of polynomials in one complex variable satisfy a recurrence relation with length growing more slowly than linearly. It is shown that (P_n) is an orthonormal basis in L²_μ for some measure μ on C, if and only if the recurrence is a 3-term relation with special coefficients. The support of μ lies on a straight line. This result is achieved by the analysis of a formally normal irreducible Hessenberg operator with only finitely many nonzero entries in every row. It generalizes the classical Favard's Theorem and the Representation Theorem.