# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Determination of all coherent pairs. (English) Zbl 0880.42012

In the study of Sobolev orthogonal polynomials, corresponding to the inner product

${〈f,g〉}_{S}={\int }_{a}^{b}fgd{{\Psi }}_{0}+\lambda {\int }_{a}^{b}{f}^{\text{'}}{g}^{\text{'}}d{{\Psi }}_{1},$

where ${{\Psi }}_{0}$ and ${{\Psi }}_{1}$ are distribution functions and $\lambda \ge 0$, an essential role has been played by the concept of coherence. If $\left\{{P}_{n}\right\}$ and $\left\{{T}_{n}\right\}$ are monic orthogonal polynomial sequences corresponding to ${{\Psi }}_{0}$ and ${{\Psi }}_{1}$, respectively, then $\left({{\Psi }}_{0},{{\Psi }}_{1}\right)$ is a coherent pair if

${T}_{n}=\frac{{P}_{n+1}^{\text{'}}}{n+1}-{\sigma }_{n}\frac{{P}_{n}^{\text{'}}}{n},\phantom{\rule{1.em}{0ex}}{\sigma }_{n}=\text{const.}\phantom{\rule{4.pt}{0ex}}\ne 0·\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 $\left({u}_{0},{u}_{1}\right)$ 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 $n-1$.

##### MSC:
 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type