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Determination of all coherent pairs. (English) Zbl 0880.42012
In the study of Sobolev orthogonal polynomials, corresponding to the inner product $$ \langle f, g \rangle_S=\int_a^b f g d\Psi_0 + \lambda \int_a^b f' g' d\Psi_1 , $$ where $\Psi_0$ and $\Psi_1$ are distribution functions and $\lambda \geq 0$, an essential role has been played by the concept of coherence. If $\{P_n\}$ and $\{T_n\}$ are monic orthogonal polynomial sequences corresponding to $\Psi_0$ and $\Psi_1$, respectively, then $(\Psi_0,\Psi_1)$ is a coherent pair if $$ T_n = \frac{P_{n+1}'}{n+1} - \sigma_n \frac{P_{n}'}{n} , \quad \sigma_n=\text{const. } \neq 0 . \tag{*} $$ This concept can be extended to quasi-definite linear functionals $u_0$ and $u_1$. When $u_0$ and $u_1$ are coherent, the Sobolev polynomial sequence has a structure suitable for further study. Nevertheless, it is important to know all the coherent pairs precisely, and this is the aim of this paper. The author proves that if $(u_0, u_1)$ is a coherent pair, then at least one of the functionals is classical (Hermite, Laguerre, Jacobi or Bessel). Moreover, all the coherent pairs are given explicitly. Analogous results are established for symmetric coherent pairs, when the subindices involved in the r.h.s. of (*) are $n+1$ and $n-1$.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
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