Asymptotic behaviour of Laguerre–Sobolev-type orthogonal polynomials. A nondiagonal case

In this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-type inner product 〈 p , q 〉 S = ∫ 0 ∞ p ( x ) q ( x ) x α e − x d x + P ( 0 ) t A Q ( 0 ) , α > − 1 , where p and q are polynomials with real coefficients, A = ( M 0 λ λ M 1 ) , P ( 0 ) = ( p ( 0 ) p ′ ( 0 ) ) , Q ( 0 ) = ( q ( 0 ) q ′ ( 0 ) ) , and A is a positive semidefinite matrix. l focus our attention on their outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler–Heine formula for the rescaled polynomials.