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On Laguerre-Sobolev type orthogonal polynomials: zeros and electrostatic interpretation. (English) Zbl 1291.42022
The author considers the Laguerre-Sobolev type polynomials, denoted by \(\{ \tilde{L_{n}^{\alpha}}(x) \}_{n}\), which are orthogonal with respect to the inner product
(1) \(\langle p, q \rangle = \int_{0}^{\infty} p(x)q(x) e^{-x} x ^{\alpha} dx + Mp(a)q(a) + N p'(a)q'(a), \)
where \(M,N \geq 0\), \(\alpha >-1\) and \(a<0\), aiming to study the interlacing of the zeros of elements of relevant polynomial sequences of this class. The initial survey on the subject is followed by the proof of an interlacing property fulfilled by the zeros of \(\tilde{L_{n}^{\alpha}}(x)\) and the zeros of the classical Laguerre polynomial of the same degree \(L_{n}^{\alpha}(x)\), when all the zeros of \(\tilde{L_{n}^{\alpha}}(x)\) are greater than \(a\). Further interlacing features are indicated concerning the zeros of the polynomial sets orthogonal with respect to (1) with \(M=0\) and \(N=0\), respectively. Also, a differential identity is deduced, together with an electrostatic interpretation of the zeros of \(\tilde{L_{n}^{\alpha}}(x)\) following, in particular, references [M. E. H. Ismail, Pac. J. Math. 193, No. 2, 355–369 (2000; Zbl 1011.33011); Numer. Funct. Anal. Optim. 21, No. 1–2, 191–204 (2000; Zbl 0981.42015)]. The final section provides a few numerical examples using Matlab software.
MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Software:
Matlab
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