On Sobolev Orthogonal Polynomials with Coherent Pairs: The Laguerre Case, Type 1

We study the asymptotic properties of orthogonal polynomials with Sobolev inner product f,g:sHbf x.g x. dm x.qlHbfX x.gX x.dn x.. a a The pair dm, dn4is called a coherent pair if there exists nonzero constants Dn such that PnXq1 x. PnX x. Qn x.s nq1 qDn n , nG1, where Pn x. and Qn x. are the nth monic orthogonal polynomials with respect to dm and dn , respectively. One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types in each case. We consider the nth root asymptotics and the zero distribution for the Laguerre case, type 1.