On series of orthogonal polynomials and systems of classical type polynomials - Zagorodnyuk, S. M. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.587If $\displaystyle\sum olimits_{k=0}^∞ c_k g_k(x),$ is a formal series of orthonormal polynomials $g_k(x)$ on the real line that has positive coefficients $c_k,$ then its partial sums $u_n(x)$ are associated with Jacobi type pencils.Therefore, they possess a recurrence relation and special orthonormality conditions.The cases where $g_k(x)$ are Jacobi or Laguerre polynomials will be of a special interest.For a suitable choice of parameters~$c_k,$ the partial sums $u_n(x)$ are Sobolev orthogonal polynomials with a $(3× 3)$ matrix measure.A~further selection of parameters gives differential equations for $u_n.$In this case, polynomials $u_n(x)$ are solutions to generalized eigenvalue problems both in $x$ and in $n.$