Varying discrete Laguerre-Sobolev orthogonal polynomials: Asymptotic behavior and zeros

We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler-Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give us an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other special functions. We generalize some results appeared very recently in the literature for both the varying and non-varying cases.The author FM is partially supported by Dirección General de Investigación, Ministerio de Economía y Competitividad Innovación of Spain, Grant MTM2012 36732 C03 01. The author JJMB is partially supported by Dirección General de Inves tigación, Ministerio de Ciencia e Innovación of Spain and European Regional Development Found, Grant MTM2011 28952 C02 01, and Junta de Andalucía, Research Group FQM 0229 (belonging to Campus of International Excellence CEI MAR), and projects P09 FQM 4643 and P11 FQM 7276

On asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

AbstractIn this paper, we consider the asymptotic behavior of the sequence of monic polynomials orthogonal with respect to the Sobolev inner product〈p,q〉S=∫0∞p(x)q(x)dμ+Mp(m)(ζ)q(m)(ζ),where ζ<0, M⩾0 and dμ=e−xxαdx. We study the outer relative asymptotics of these polynomials with respect to the classical Laguerre polynomials, and we deduce a Mehler–Heine type formula and a Plancherel–Rotach type formula for the rescaled polynomials

Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.Comment: Changed content; 34 pages, 41 reference

Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure for x in [0,?), ? > ?1, a free parameter N and a shift c<0. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as n tends to infinity

On analytic properties of Meixner-Sobolev orthogonal polynomials of higher order difference operators

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product $$\left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr T^{j}g(\alpha),$$ where ${\bf u}^{\tt M}$ is the Meixner linear operator, $\lambda\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $\alpha \leq 0$, and $\mathscr T$ is the forward difference operator $\Delta$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the Mehler-Heine type formula for the $\alpha\le 0$ case