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The holonomic equation of the Laguerre-Sobolev-type orthogonal polynomials: a non-diagonal case. (English) Zbl 1219.42012

The authors consider the Sobolev-type inner product
\[ \langle p,q \rangle_{S}=\int_{0}^{\infty}p(x)q(x)x^{\alpha}e^{-x}\,dx+\mathbb{P}(0)^{t}A\mathbb{Q}(0), \quad\alpha>-1, \]
where \(p\) and \(q\) are polynomials with real coefficients,
\[ A=\left(\begin{matrix} M_{0} & \lambda\\ \lambda & M_{1} \end{matrix}\right),\qquad \mathbb{P}(0)=\binom{p(0)}{p'(0)},\qquad \mathbb{Q}(0)=\binom{q(0)}{q'(0)}, \]
and \(A\) is a positive semi-definite matrix.
The authors consider a multiplication operator that is symmetric with respect to the above inner product. As a consequence, they prove that the sequence of monic polynomials orthogonal with respect to the above inner product satisfies a five-term recurrence relation. They also obtain raising and lowering operators associated with them. As a consequence, a holonomic equation satisfied by these polynomials is given.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C47 Other special orthogonal polynomials and functions
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References:

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