On series of orthogonal polynomials and systems of classical type polynomials (Q2058600)

If \(\sum_{k=0}^{\infty} c_{k} g_{k}(x)\) is a formal series of orthonormal polynomials \(g_{k}(x)\) on the real line with 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 are of especial interest. In the present paper, the authors construct examples of polynomials \(u_{n}(x)\) leading to Sobolev orthogonal polynomials of the classical type and prove (in Theorem 1) that they satisfy a recurrence relation of the form \[ \begin{aligned} \gamma_{n-2} p_{n-2}(\lambda) &+\left(\beta_{n-1}-\lambda a_{n-1}\right) p_{n-1}(\lambda) \\ &+\left(\alpha_{n}-\lambda b_{n}\right) p_{n}(\lambda)+\left(\beta_{n}-\lambda a_{n}\right) p_{n+1}(\lambda)+\gamma_{n} p_{n+2}(\lambda)=0, \quad n \in \mathbb{Z}_{+}, \end{aligned} \] where \[ p_{-2}(\lambda)=p_{-1}(\lambda)=0 \quad \text { and } \quad \gamma_{-2}=\gamma_{-1}=a_{-1}=\beta_{-1}=0 . \]