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On orthogonal polynomials with respect to a class of differential operators. (English) Zbl 1298.33017

The author considers orthogonal polynomials with respect to linear differential operators of the form \[ \mathcal L^{(M)} := \sum_{k=0}^M \rho_k (z) \frac{d^k}{dz^k}, \] where \(\{\rho_k\}_{k=0}^M\) are complex polynomials with \(\deg \rho_k \leq k\), \(0\leq k \leq M\), and equality for at least one index. Such a differential operator is called exactly solvable if \(\mathcal L^{(M)} (\mathbb{P}_n) \subseteq\mathbb{P}_n\), \(\forall n\in \mathbb N\), with equality for at least one \(n\). Here, \(\mathbb{P}_n\) denotes the linear space of all polynomials of degree at most \(n\). Connections between this type of orthogonality and inner products are presented and the exactly solvable operators for which this concept reduces to an inner product are classified. Necessary and sufficient conditions for the normality of an index \(n\) are derived. The existence of infinite sequences of polynomials \(\{Q_n\}_{n=m+1}^\infty\) for some nonnegative \(m\in\mathbb Z_0^+\), with \(\deg Q_n = n\), in terms of a linear system of difference equations with varying coefficients is investigated. In addition, the location of the zeros of the polynomials \(\{Q_n\}\) is studied.

MSC:

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