Herbert Dueñas

The Laguerre–Sobolev-type orthogonal polynomials

Abstract In this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-type inner product 〈 p , q 〉 S = ∫ 0 ∞ p ( x ) q ( x ) x α e − x d x + N p ( j ) ( 0 ) q ( j ) ( 0 ) , where N ∈ R + and j ∈ N . We will focus our attention on the outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler–Heine formula for the rescaled polynomials.

Asymptotic properties of Laguerre---Sobolev type orthogonal polynomials

AbstractIn this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product

Jacobi–Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation – a non-diagonal case

\left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1

The Laguerre-Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation

where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler–Heine formula as well as a Plancherel–Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N.

Jacobi-T ype orthogonal polynomials: holonomic equation and electrostatic interpretation

In this paper, we consider sequences of monic polynomials orthogonal with respect to an inner product where M∈ℝ+, and a∈ℝ−. We focus our attention on the representation of these polynomials in terms of the standard Laguerre polynomials as well as hypergeometric functions. The lowering and raising operators associated with these polynomials are obtained. The distribution of their zeros is analysed in terms of their dependence of M. Finally, some outer asymptotic properties of such orthogonal polynomials are discussed.

Asymptotic properties of LaguerreSobolev type orthogonal polynomials

Consider the Sobolev-type inner product where p and q are polynomials with real coefficients, α, β>−1, ℙ(x)=(p(x), p′(x)) t , and is a positive semidefinite matrix, with M 0, M 1≥0, and λ∈ℝ. We obtain an expression for the family of polynomials , orthogonal with respect to the above inner product, a connection formula that relates with some family of Jacobi polynomials and the holonomic equation that they satisfy, as well as an electrostatic interpretation of their zeros.

Comportamiento asintótico de polinomios ortogonales en varias variables sobre la bola unidad

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived.