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Companion linear functionals and Sobolev inner products: a case study. (English) Zbl 1087.42020
Let $\Bbb P$ be the space of all polynomials with real coefficients. A linear functional $U:\Bbb P\to\Bbb R$ is said to be semiclassical if there exists two polynomials $\varphi$ and $\psi$ such that the following distributional Pearson equation holds $$ D(\varphi U)=\psi U \tag 1$$ where linear functionals $\psi U$ and $D(\varphi U)$ are defined as $$ \langle \psi U,p \rangle:=\langle U,\psi p \rangle, \quad \langle D(\varphi U),p \rangle=-\langle U,\varphi p' \rangle,\quad p\in\Bbb P. $$ A semiclassical linear functional $U$ is said to be of class $s$ if $$ \min_\Omega\max(\deg \varphi-2, \deg \psi-1)=s $$ where $\Omega=\Omega(U)$ is the set of all possible pairs of polynomials $\varphi,\psi$ from Pearson equation (1). Let $V,U$ be two quasi-definite linear functionals on $\Bbb P$ and let $\{R_n\},\{P_n\}$ be corresponding sequences of monic orthogonal polynomials, i.e., $$ \deg R_n=\deg P_n=n \quad\text{and}\quad \frac1{K_n}\langle U,R_nR_m \rangle=\frac1{\Gamma_n}\langle V,P_nP_m \rangle=\delta_{n,m} $$ where $K_n\ne0\ne\Gamma_n$ and $\delta_{n,m}$ is Kronecker delta. The following problem is completely solved: “Describe the pairs of quasi-definite linear functionals $U,V$ for which the differential relation $$ \frac{R'_{n+1}(x)}{n+1}+b_n\frac{R'_n(x)}{n}=P_n(x)+a_nP_{n-1}(x) \tag 2$$ holds for all positive integer $n$ with $ b_n\ne0$.” The authors prove that at least one of the functionals $U,V$ is semiclassical of class at most 1 if the differential relation (2) holds, and this fact gives a clue to the structure of pairs $(U,V)$.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
33C47Other special orthogonal polynomials and functions
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