What is beyond coherent pairs of orthogonal polynomials? (English) Zbl 0855.42016
Two quasi-definite linear functionals \(u_0\) and \(u_1\) constitute a coherent pair when the corresponding monic orthogonal polynomial systems, \(\{P_n\}_n\) and \(\{T_n\}_n\), satisfy
\[
T_n (x) = {P_{n + 1}' (x) \over n + 1} - \sigma_n {P_n' (x) \over n},\;n \geq 1, \tag{*}
\]
where \(\sigma_n\) are nonzero real numbers. The authors fix one of the monic orthogonal polynomial systems \((P_n\) or \(T_n)\) and study conditions on \(\{\sigma_n\}\) that make the second sequence, given by (*), to be orthogonal.
Furthermore, the symmetric coherence of \(u_0\) and \(u_1\) is defined by a relation analogous to (*), but where \(P_{n + 1}\) and \(P_{n - 1}\) are involved. The construction of symmetrically coherent pairs of functionals from the usual coherence is also investigated.
Furthermore, the symmetric coherence of \(u_0\) and \(u_1\) is defined by a relation analogous to (*), but where \(P_{n + 1}\) and \(P_{n - 1}\) are involved. The construction of symmetrically coherent pairs of functionals from the usual coherence is also investigated.
Reviewer: A.Martínez Finkelshtein (Almeria)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
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