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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.

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