Second degree classical forms (Q1384931)

A linear functional (form) \(u\) over the space of polynomials is regular if it admits a sequence of polynomials, orthogonal with respect to \(u\). Additionally, \(u\) is a second degree form if its Stieltjes transform \[ S(z)=\sum_{n=0}^\infty {u(z^n) \over z^{n+1}} \] is an algebraic function of degree \(2\) (in other words, \(S\) is a solution of a quadratic equation with polynomial coefficients in \(z\)). The authors show that among the classical forms (corresponding to the case when differentiation preserves orthogonality with respect to \(u\)) the only second degree form is the Jacobi one, with a suitable selection of parameters.