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On orthogonal polynomials with respect to certain discrete Sobolev inner product. (English) Zbl 1259.33023
Polynomials orthogonal with respect to an inner product $\langle f,g\rangle = \int_E \omega (x) f(x) g(x)\, dx + M f(\xi) g(\xi) + N f' (\xi) g' (\xi), \quad \tag{1}$ where $$\xi$$ is a real number and $$d\mu$$ is a positive Borel measure supported on an infinite subset $$E$$ of the real line, have been considered by several authors and are known in the literature as Sobolev-type or discrete Sobolev orthogonal polynomials.
In this paper, the authors consider the discrete Sobolev polynomials $$\{\widehat{S}_n\}_{n \geq0}$$ orthogonal with respect to $$(1)$$, where $$E=[0, +\infty)$$ and $$\xi < 0$$. They show that these polynomials can be expressed as $\widehat{S}_n(x) = \widehat{P}_n (x) + A_{n,1} (x-\xi)\widehat{P}_{n-1}^{[2]} (x) + A_{n,2} (x-\xi)^2 \widehat{P}_{n-2}^{[4]} (x),$ where $$\{\widehat{P}_n \}_{n \geq 0}$$ and $$\{ \widehat{P}_n^{[k]} \}_{n \geq 0}$$, $$k \in \mathbb{N}$$, are the sequences of monic polynomials orthogonal with respect to the weight functions $$\omega(\cdot)$$ and $$(\cdot -\xi)^k \omega(\cdot)$$, respectively.
In Section 3, the location of the zeros of discrete Sobolev orthogonal polynomials $$\widehat{S}_n$$ is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function $$\omega.$$ Here is the theorem.
Theorem 3. Denote by $$\nu_{r,n}$$, $$r=1,2, \dotsc, n$$, the zeros of $$\widehat{S}_n (x)$$ in increasing order. Suppose that $$\nu_{1,n} < \xi$$. Then $$2\xi - x^{[2]}_{1, n-1} < \nu_{1,n} < \xi$$ and $\xi < \nu_{2,n} < x^{[2]}_{1, n-1} < \dotsb < \nu_{n,n} < x^{[2]}_{n-1, n-1}.$ Moreover, the behavior of the coefficients $$A_{n,1}$$ and $$A_{n,2}$$ is studied in more detail. In particular, when $$\omega$$ is the Laguerre weight, they obtain some asymptotic properties for the sequence of discrete Laguerre-Sobolov orthogonal polynomials. More precisely, the authors obtain outer relative asymptotics, a Mehler-Heine formula and the Plancherel-Rotach outer asymptotics for such orthogonal polynomials.

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