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Álvarez-Nodarse, R.; Petronilho, J.

On the Krall-type discrete polynomials. (English) Zbl 1051.33006

J. Math. Anal. Appl. 295, No. 1, 55-69 (2004).
Let \(\mathbf{u}\) be a quasi-definite linear functional in the vector space \(\mathbb{P}\) of polynomials with complex coefficients. Then there exists a sequence of monic polynomials (\(P_{n})\) with \(\deg P_{n}=n\), such that \(\langle \mathbf{u},\,P_{n}P_{m}\rangle=k_{n}\,\delta_{n,\,m}\), \(k_{n}\neq 0\), \(n,m=0,1,2,\ldots\). Considering perturbations of the functional \(\mathbf{u}\) of the form \[ \tilde{\mathbf{u}}=\mathbf{u}+\sum_{i=1}^{M}A_{i}\,\delta(x-a_{i}), \] where \((A_{i})_{i=1}^{M}\) are nonzero real numbers and \(\delta(x-y)\) is the Dirac linear functional defined by \(\langle \delta(x-y),p(x)\rangle=p(y), \forall p\in \mathbb{P}\), one obtains the so-called Krall-type orthogonal polynomials. In this paper the authors study the case where the functional \(\mathbf{u}\) in (1) is a semiclassical discrete or \(q\)-discrete functional making a special emphasis in the case where \(\mathbf{u}\) is a classical discrete or \(q\)-classical functional. The authors present a unified theory for studying the Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of \(q\)-Krall polynomials are considered in detail.
Reviewer: Stamatis Koumandos (Nicosia)

Cited in 1 Review
Cited in 9 Documents

MSC:

33C47 Other special orthogonal polynomials and functions
33E30 Other functions coming from differential, difference and integral equations
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

quasi-definite linear functionals; classical discrete polynomials; \(q\)-polynomials; Krall-type polynomials; addition of delta Dirac masses
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References:

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