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Companion linear functionals and Sobolev inner products: a case study. (English) Zbl 1087.42020

Let \(\mathbb P\) be the space of all polynomials with real coefficients. A linear functional \(U:\mathbb P\to\mathbb R\) 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 \[ \langle \psi U,p \rangle:=\langle U,\psi p \rangle, \quad \langle D(\varphi U),p \rangle=-\langle U,\varphi p' \rangle,\quad p\in\mathbb P. \] 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 \(\mathbb P\) 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 \(K_n\neq0\neq\Gamma_n\) 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 \( b_n\neq0\).” 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 \((U,V)\).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
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