On Laguerre-Sobolev type orthogonal polynomials, zeros and electrostatic interpretation

Luis Alejandro Molano Molano

Abstract


We study the sequence of monic polynomials orthogonal with respect to inner product
$$
\langle p,q\rangle = \int_0^\infty p(x)q(x)e^{-x}x^{\alpha}\;dx
+Mp(\zeta )q(\zeta )+Np^\prime(\zeta )q^\prime(\zeta),
$$
where \(\alpha >-1\), \(M\geq 0\), \(N\geq 0\), \(\zeta < 0\), and \(p\) and \(q\) are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros.

doi:10.1017/S1446181113000308

Keywords


orthogonal polynomials, Sobolev-type inner products, Laguerre polynomials, zeros, electrostatic interpretation



DOI: http://dx.doi.org/10.21914/anziamj.v55i0.6673



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.