Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives

Marcellán, Francisco; Rafaeli, Fernando

doi:10.1090/S0002-9939-2011-10806-2

Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives
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by Francisco Marcellán and Fernando R. Rafaeli PDF

Proc. Amer. Math. Soc. 139 (2011), 3929-3936 Request permission

Abstract:

In this paper we analyze the location of the zeros of polynomials orthogonal with respect to the inner product \begin{equation} \langle p,q\rangle = \displaystyle \int _{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{(j)}(0)q^{(j)}(0), \end{equation} where $\alpha >-1$, $N\geq 0,$ and $j\in \mathbb {N}.$ In particular, we focus our attention on their interlacing properties with respect to the zeros of Laguerre polynomials as well as on the monotonicity of each individual zero in terms of the mass $N.$ Finally, we give necessary and sufficient conditions in terms of $N$ in order for the least zero of any Laguerre-Sobolev-type orthogonal polynomial to be negative.

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