Companion orthogonal polynomials: some applications. (English) Zbl 1003.42014
The authors consider two distributions (positive Borel measures) on \(\mathbb R\), \(\phi_0\) and \(\phi_1\), symmetric with respect to the origin, and such that one is a rational modification of the other, namely
\[
d\phi_1(x)={c \over 1+qx^2} d\phi_0(x), \qquad \int d\phi_1= \int d\phi_0.
\]
The paper extends the previous results by one of the authors [A. Sri Ranga, “Companion orthogonal polynomials”, J. Comput. Appl. Math. 75, No. 1, 23–33 (1996; Zbl 0861.42017)] by showing that for all the admissible values of the parameter \(q\) the coefficient of the three-term recurrence relation satisfied by the polynomials orthogonal with respect to \(\phi_0\) and to \(\phi_1\), can be explicitly expressed in terms of a sequence \(\ell_n=\ell_n(q)\).
This result is applied to the study of Sobolev orthogonal polynomials corresponding to the so-called symmetric coherent pairs, allowing to derive new algebraic relations for these polynomials. Further applications deal with measures generating orthogonal polynomial sequences with double periodic recurrence coefficients.
This result is applied to the study of Sobolev orthogonal polynomials corresponding to the so-called symmetric coherent pairs, allowing to derive new algebraic relations for these polynomials. Further applications deal with measures generating orthogonal polynomial sequences with double periodic recurrence coefficients.
Reviewer: Andrei Martínez Finkelshtein (Almeria)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
orthogonal polynomials; three-term recurrence relation; Sobolev orthogonal polynomials; measuresCitations:
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