Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

Abstract

We investigate algebraic and analytic properties of sequences of polynomials orthogonal with respect to the Sobolev type inner product $$\begin{aligned} \langle f,g\rangle = \int \!\!f(x)g(x)d\mu (x)+ \sum _{k=1}^{K}\sum _{i=0}^{N_{k}}M_{k,i}f^{(i)}(b_{k})g^{(i)}(b_{k}), \end{aligned}$$ where $$\mu $$ is a finite positive Borel measure belonging to the Nevai class, the mass points $$b_{k}$$ are located outside the support of $$\mu $$ , and $$M_{k,i}$$ are complex numbers such that $$M_{k,N_{k}}\ne 0$$ . First, we study the existence as well as recurrence relations for such polynomials. When the values $$M_{k,i}$$ are nonnegative real numbers, we can deduce the coefficients of the recurrence relation in terms of the connection coefficients for the sequences of polynomials orthogonal with respect to the Sobolev type inner product and those orthogonal with respect to the measure $$\mu $$ . The matrix of a symmetric multiplication operator in terms of the above sequence of Sobolev type orthogonal polynomials is obtained from the Jacobi matrix associated with the measure $$\mu $$ . Finally, we focus our attention on some outer relative asymptotics of such polynomials, which are deduced by using the above connection formulas.