On polar Legendre polynomials

10 pages, no figures.-- MSC2000 codes: Primary 42C05; Secondary 33C25.-- ArXiv pre-print available at: http://arxiv.org/abs/0709.4537Accepted in Rocky Mountain Journal of Mathematics.We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product.Research by first author (H.P.) was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología de España, under grant MTM2006-13000-C03-02, by Comunidad de Madrid-Universidad Carlos III de Madrid, under grant CCG06-UC3M/EST-0690 and by Centro de Investigación Matemática de Canarias (CIMAC). Research by second author (J.Y.B.) was supported by CNPq-TWAS. Research by third author (W.U.) was partially supported by Centro de Investigación Matemática de Canarias (CIMAC).En prens

Iterated integrals of Jacobi polynomials

Let P(α,β)n be the n-th monic Jacobi polynomial with α,β>−1. Given m numbers ω1,…,ωm∈C∖[−1,1], let Ωm=(ω1,…,ωm) and P(α,β)n,m,Ωm be the m-th iterated integral of (n+m)!n!P(α,β)n normalized by the conditions dkP(α,β)n,m,Ωmdzk(ωm−k)=0, for k=0,1,…,m−1. The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials {P(α,β)n,m,Ωm}n. In particular, we obtain the relative asymptotic for the ratio of the sequences {P(α,β)n,m,Ωm}n and {P(α,β)n}n. We prove that the zeros of these polynomials accumulate on a suitable ellipse.The research of H. Pijeira was supported by research Grant MTM2015-65888-C4-2-P Ministerio de Economía y Competitividad of Spain

Differential orthogonality: Laguerre and Hermite cases with applications

Let μ be a finite positive Borel measure supported on R , L[f]=xf′′+(α+1−x)f′ with α>−1 , or L[f]=12f′′−xf′ , and m a natural number. We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials {Qn}n>m that satisfy the orthogonality relations ∫L[Qn](x)xkdμ(x)=0for all0≤k≤n−1. Turn MathJax off We also provide a fluid dynamics model for the zeros of these polynomials.The authors thank the comments and suggestions made by the referees which helped improve the manuscript. First author’s research was partially supported by FAPESP of Brazil, under grant 2012/21042-0. First and second authors’ research was partially supported by Ministerio de Economía y Competitividad of Spain, under grant MTM2012-36732-C03-01

Electrostatic models for zeros of Laguerre-Sobolev polynomials

Let {$\{S_n\}_{n\geqslant 0}$} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product $$\langle f,g\rangle_S =\!\int_{0}^{+\infty}\! f(x) g(x)x^{\alpha}e^{-x}dx+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j),$$ where $\lambda_{j,k}\geqslant 0$, $\alpha >-1$ and $c_i \in (-\infty, 0)$ for $i=1,2,\dots,N$. We provide a formula that relates the Laguerre-Sobolev polynomials $S_n$ to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence $\{S_n\}_{n\geqslant 0}$ and a second-order differential equation with polynomial coefficients for $\{S_n\}_{n\geqslant 0}$. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre-Sobolev polynomials. Some examples are given where this condition is either satisfied or not.Comment: arXiv admin note: substantial text overlap with arXiv:2308.0617

Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation

It is very well known that a sequence of polynomials {Q(n)(x)}(n=0)(infinity) orthogonal with respect to a Sobolev discrete inner product (s) = integral(I)fg d mu + lambda f(-1)(0)g'(0); lambda is an element of R+; where mu is a finite Borel measure and I is an interval of the real line, satisfies a five- term recurrence relation. In this contribution we study other three families of polynomials which are linearly independent solutions of such a five- term linear difference equation and, as a consequence, we obtain a polynomial basis of such a linear space. They constitute the analogue of the associated polynomials in the standard case. Their explicit expression in terms of {Q(n)(x)}(n=0)(infinity) using an integral representation is given. Finally, an application of these polynomials in rational approximation is shown

Hessenberg-Sobolev matrices and Favard type theorem

We study the relation between certain non-degenerate lower Hessenberg infinite matrices G and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for Sobolev orthogonality. We characterize the structure of the matrix G and the associated matrix of formal moments MG in terms of certain matrix operators.The research of I. Pérez-Yzquierdo was partially supported by Fondo Nacional de Innovación y Desarrollo Científico y Tecnológico (FONDOCYT), Dominican Republic, under grant 2016-2017-080 No. 013-2018. The authors thank the reviewers for theirs constructive comments and suggestions that helped to improve the clarity of this manuscript. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature

Asymptotic zero distribution for a class of extremal polynomials

We consider extremal polynomials with respect to a Sobolev-type p-norm, with 1and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the nth root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.A.D.G. was supported by the Research Fellowship Program, Ministerio de Economía, Industria y Competitividad of Spain, under grant BES-2016-076613. The authors G.L.L. and H.P.C. were supported by the Ministerio de Economía, Industria y Competitividad of Spain, under grant MTM2015-65888-C4-2-P