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

Abstract

Define P n ( x ) P_n(x) and Q n ( x ) Q_n(x) as the n n th monic orthogonal polynomials with respect to d μ d\mu and d ν d\nu respectively. The pair { d μ , d ν } \{d\mu ,d\nu \} is called a coherent pair if there exist non-zero constants D n D_n such that \[ Q n ( x ) = P n + 1 ′ ( x ) n + 1 + D n P n ′ ( x ) n , n ≥ 1. Q_n(x)=\frac {P_{n+1}^\prime (x)}{n+1}+D_n\frac {P_n^\prime (x)}{n},\qquad n\ge 1. \] 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 \[ ⟨ f , g ⟩ = ∫ a b f ( x ) g ( x ) d μ ( x ) + λ ∫ a b f ′ ( x ) g ′ ( x ) d ν ( x ) \langle f,g\rangle =\int _a^b f(x)g(x)d\mu (x)+\lambda \int _a^b f’(x)g’(x)d\nu (x) \] for the coherent pair { d μ , d ν } \{d\mu ,d\nu \} : the Jacobi case, type 1.