On the Krall-type polynomials
Let \(u\) be a quasi-definite linear functional in the vector space \(\mathbb{P}\) of polynomials with complex coefficients. Perturbations of such functionals \(u\) via the addition of Dirac delta functionals \(\delta\) and their derivatives \(\delta^{\prime}\) have been the subject of extensive study in recent years. In this paper, the authors study perturbed linear functionals of the form \(\tilde{u}=u+ A\delta(x-a)-B\delta^{\prime}(x-b),\;\;a\neq b\). In particular, they give necessary and sufficient conditions for the quasi-definiteness of such a \(\tilde{u}\). In this case, the coefficients of the three-term recurrence relation that the corresponding sequence of monic orthogonal polynomials satisfies is obtained. In the case of a semiclassical linear functional \(u\), it is shown that the corresponding functional \(\tilde{u}\) is also semiclassical. This can be read in terms of the holonomic second-order linear differential equation that such polynomials satisfy. The explicit expression of their coefficients is given. Information on the location of zeros of these polynomials is also derived. The paper also provides background material on the subject as well as an extensive list of relevant papers.
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