Rocky Mountain Journal of Mathematics

On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

J.F. Sánchez-Lara

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The discrete part of the discrete-continuous orthogonality \[ \mathscr {B}(f,g)=\mathscr {B}_d( f,g)+\mathscr {B}_c(f^{(N)},g^{(N)}), \] is studied for families of classical orthogonal polynomials such that the associated three-term recurrence relation \[ xp_n=p_{n+1}+\beta _np_n+ \gamma _n p_{n-1}, \] presents one vanishing coefficient $\gamma _n$, as in the case of Laguerre polynomials $L_n^{(-N)}$, Jacobi polynomials $P_n^{(-N,\beta )}$ and Gegenbauer polynomials $C_n^{(-N+1/2)}$ with $N\in \mathbb {N}$. It is shown that the discrete bilinear functional $\mathscr {B}_d$ can be replaced by a linear functional, $\mathscr {L}$, or by another bilinear functional related with $\mathscr {L}$, which allows us to reformulate the orthogonality in a much simpler way in the case of Laguerre polynomials and in a totally explicit form in the case of Jacobi and Gegenbauer polynomials.

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Rocky Mountain J. Math., Volume 47, Number 1 (2017), 267-288.

First available in Project Euclid: 3 March 2017

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Zentralblatt MATH identifier

Primary: 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Classical orthogonal polynomials Sobolev orthogonality


Sánchez-Lara, J.F. On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters. Rocky Mountain J. Math. 47 (2017), no. 1, 267--288. doi:10.1216/RMJ-2017-47-1-267.

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