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)xαe−xdx+P(0)tAQ(0),α>−1, where pp and qq are polynomials with real coefficients, A=(M0λλM1),P(0)=(p(0)p′(0)),Q(0)=(q(0)q′(0)), and AA is a positive semidefinite matrix.We will 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.