A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind

We show that the only orthogonal polynomials with a generating function of the form $F(xz-\alpha z^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(xw(z)-R(z))$.