The D\(_{u}\)-classical orthogonal polynomials. (English) Zbl 1408.33018
Summary: The \(\mathbf D_u\)-classical orthogonal polynomial sequences are defined through the \(\mathbf D_u\)-Hahn’s property: sequences that are orthogonal together with their \(\mathbf D_u\)-first derivative, where \(\mathbf D_u(p) = p^\prime + u\theta_0p\), for all \(p \in \mathbb C[X]\). We characterize them by means of a functional equation, a \(\mathbf D_u\)-second order linear differential equation, the first and the second structure relations. A \(\mathbf D_u\)-classical orthogonal sequence is especially a \(\mathbf D\)-Laguerre-Hahn sequence of class less than or equal to two. A complete classification of the \(\mathbf D_u\)-classical sequences is obtained. The functional equation coefficients, the structure relations coefficients, the three-term recurrence relation coefficients and the class are whenever given.
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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