On series of orthogonal polynomials and systems of classical type polynomials - Zagorodnyuk, S. M. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.587 If 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.