2020/03/12 by Sergey M. Zagorodnyuk, Zagorodnyuk, Sergey M.
Mathematics · #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:42C05
paper · pdf · doi:10.48550/arxiv.2003.06040
15 pages
arxiv created 2020/03/12 · arxiv updated 2020/03/16
In this paper we study modified kernel polynomials: un(x) = ∑k=0n ck gk(x), depending on parameters ck>0, where \ gk \0^∞ are orthonormal polynomials on the real line. Besides kernel polynomials with ck = gk(t0)>0, for example, ck may be chosen to be some other solutions of the corresponding second-order difference equation of gk. It is shown that all these polynomials satisfy a 4-th order recurrence relation. The cases with gk being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters ck imply un to be Sobolev orthogonal polynomials with a (3× 3) matrix measure. Moreover, a further selection of parameters gives differential equations for un. In the latter case, polynomials un(x) are solutions to a generalized eigenvalue problems both in x and in n.