Some classical polynomials seen from another side
One characterization of the (very) classical orthogonal polynomials is that besides the sequence \(\{P_n(x)\}_{n\geq 0}\) also the sequence \(\{\frac{P_{n+1}'(x)}{n+1}\}_{n\geq 0}\) is orthogonal. The aim of this paper is to study the sequences \(\{P_n(x)\}_{n\geq 0}\) of orthogonal polynomials for which the sequence \(\{\frac{((x-c)P_n(x))'}{n+1}\}_{n\geq 0}\) is also orthogonal. The result is that these sequences have to be (very) classical, in fact they have to be Laguerre or Jacobi polynomials, where in the case of Laguerre \(c\) has to be zero and in the case of Jacobi \(c\) has to be \(\pm 1\). This result is achieved by considering the Hermite, Laguerre, Bessel and Jacobi polynomials successively.
scientific article; zbMATH DE number 1404459