Asymptotics for Sobolev orthogonal polynomials with coherent pairs: The Jacobi case, type 1

Abstract:

Define $P_n(x)$ and $Q_n(x)$ as the $n$th monic orthogonal polynomials with respect to $d\mu$ and $d\nu$ respectively. The pair $\{d\mu ,d\nu \}$ is called a coherent pair if there exist non-zero constants $D_n$ such that \[ Q_n(x)=\frac {P_{n+1}^\prime (x)}{n+1}+D_n\frac {P_n^\prime (x)}{n},\qquad n\ge 1.\] One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types for each case: type 1 and 2. We investigate the asymptotic properties and zero distribution of orthogonal polynomials with respect to Sobolev inner product \[ \langle f,g\rangle =\int _a^b f(x)g(x)d\mu (x)+\lambda \int _a^b f’(x)g’(x)d\nu (x)\] for the coherent pair $\{d\mu ,d\nu \}$: the Jacobi case, type 1.