Orthogonal polynomials and coherent pairs: The classical case
Let \(\{P_n(x)\}^\infty_{n= 0}\) and \(\{R_n(x)\}^\infty_{n= 0}\) be two sequences of orthogonal polynomials of degree equal to their index and let the parameters \(\{\sigma_n\}^\infty_{n= 0}\) satisfy \[ P_n(x)= {1\over n+ 1} R_{n+ 1}'(x)- \sigma_n R_n'(x),\quad n= 0, 1,\dots\;.\tag{1} \] The authors give a complete characterization, in the case that the given system is one of the four classical ones (Hermite, Laguerre, Jacobi, Bessel), of the conditions on the parameters \(\sigma_n\) for the two problems: P1. If \(\{R_n\}\) is a given system of orthogonal polynomials, determine all sequences of orthogonal polynomials \(\{P_n\}\) and all sequences \(\{\sigma_n\}\) for which (1) holds. P2. The same problem with the assumption that now \(\{P_n\}\) is the given system of orthogonal polynomials. The problems are motivated by the concept of coherent pairs of measures ([\textit{A. Iserles}, \textit{P. E. Koch}, \textit{S. P. Nørsett} and \textit{J. M. Sanz-Serna}, J. Approximation Theory 65, No. 2, 151-175 (1991; Zbl 0734.42016)]) in the framework of Sobolev inner products \(\langle f, g\rangle= \int_{\mathcal R} fg d\mu_0+ \lambda \int_{\mathcal R} fg d\mu_1\). The proofs are straightforward and use connecting relations between the coefficients of the three term recurrence relations for the orthogonal polynomials involved. A nice paper.
scientific article; zbMATH DE number 828208
scientific article