Asymptotics for Laguerre–Sobolev type orthogonal polynomials modified within their oscillatory regime

摘要

In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobolev inner productf,gS=∫0∞f(x)g(x)xαe-xdx+F(c)AG(c)t,α>-1,where f and g are polynomials with real coefficients, A∈R(2,2) and the vectors F(c),G(c) areA=M00N,F(c)=(f(c),f′(c))andG(c)=(g(c),g′(c)),respectively,with M,N∈R+ and the mass point c is located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of classical Laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five-term recurrence relation when the degree of the polynomials is large enough. Also, the outer relative asymptotics of the Laguerre–Sobolev type with respect to the Laguerre polynomials is analyzed.