On series of orthogonal polynomials and systems of classical type polynomials
Abstract
UDC 517.587
If $\displaystyle\sum\nolimits_{k=0}^\infty c_k g_k(x),$ 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 $(3\times 3)$ 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 $n.$
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