Sobolev-type orthogonal polynomials and their zeros

When σ is a quasi-definite moment functional on P, the space of polynomials in one variable, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) := <σ, pq> +λp^(r)(a)q^(r)(a)+µp^(s)(b)q^(s)(b), where λ, µ, a, b are complex numbers and r, s are non-negative integers. We find a necessary and sufficient condition under which there is an orthogonal polynomial system {R_n(x)}^∞_{n=0} relative to φ and discuss their algebraic properties. When σ is semi-classical, we show that {R_n(x)}^∞_{n=0} must satisfy a second order differential equation with polynomial coefficients. When σ is positive-definite and λ, µ, a, b are real, we investigate the relations between zeros of {R_n(x)}^∞_{n=0} and of the system of the orthogonal polinomiels relative to σ.