Zeros of polynomials orthogonal with respect to a signed weight

AbstractIn this paper we consider the monic polynomial sequence (Pnα,q(x)) that is orthogonal on [−1,1] with respect to the weight function x2q+1(1−x2)α(1−x),α>−1,q∈N; we obtain the coefficients of the tree-term recurrence relation(TTRR) by using a different method from the one derived in Atia et al. (2002) [2]; we prove that the interlacing property does not hold properly for (Pnα,q(x)); and we also prove that, if xn,nα+i,q+j is the largest zero of Pnα+i,q+j(x), x2n−2j,2n−2jα+j,q+j<x2n−2i,2n−2iα+i,q+i,0≤i<j≤n−1

A Survey on q-Polynomials and their Orthogonality Properties

In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson polynomials (AW)) for almost any complex value of the parameters and for all non-negative integers degrees. We state the degenerate version of Favard's theorem, which is one of the keys of the paper, that allow us to extend the orthogonality properties valid up to some integer degree N to Sobolev type orthogonality properties. We also present, following an analogous process that applied in [16], tables with the factorization and the discrete Sobolev-type orthogonality property for those families which satisfy a finite orthogonality property, i.e. it consists in sum of finite number of masspoints, such as q-Racah (qR), q-Hahn (qH), dual q-Hahn (dqH), and q-Krawtchouk polynomials (qK), among others. -- [16] R. S. Costas-Santos and J. F. Sanchez-Lara. Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comp. Appl. Math., 225(2) (2009), 440-451Comment: 3 Figures, 3 tables, in a 22 pages manuscrip

The semiclassical--Sobolev orthogonal polynomials: a general approach

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$_S= +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}>,$$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theor

Extensions of discrete classical orthogonal polynomials beyond the orthogonality

It is well known that the family of Hahn polynomials $\{h_n^{\alpha,\beta}(x;N)\}_{n\ge 0}$ is orthogonal with respect to a certain weight function up to $N$. In this paper we present a factorization for Hahn polynomials for a degree higher than $N$ and we prove that these polynomials can be characterized by a $\Delta$-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials and give the limit relations between them for all n\in \XX N_0. Furthermore, in order to get this results for the Krawtchouk polynomials we will get a more general property of orthogonality for Meixner polynomials.Comment: 2 figures, 20 page

Limit relations between qq-Krall type orthogonal polynomials

In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a $q$-classical linear functional via the addition of one or two mass points. The limit relations between the $q$-Krall type modification of big $q$-Jacobi, little $q$-Jacobi, big $q$-Laguerre, and other families of the $q$-Hahn tableau are established.Comment: 19 Pages, 3 tables, 1 figur

The Hahn Quantum Variational Calculus

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted: 18/June/2010; for publication in Journal of Optimization Theory and Application

Disponibilidade e qualidade do capim-elefante com e sem irrigação adubado com nitrogênio e potássio na estação seca Forage yield and quality of elephantgrass response to irrigation and nitrogen and potassium fertilization in the dry season

Objetivou-se com este trabalho avaliar os efeitos da irrigação e da adubação com N e K sobre a disponibilidade e a qualidade da matéria seca em pastagem de capim-elefante cv. Napier adubada com N e K nas dosagens de 100:80; 200:160; 300:240 e 400:320 kg.ha-1.ano-1 de N e K2O, respectivamente, no período seco do ano. Foram conduzidos dois experimentos (área irrigada - AI e não-irrigada - ANI), segundo delineamento de blocos casualizados com três repetições, em que a unidade experimental consistiu de parcelas (piquetes) de 300 m², nas quais foram aplicados os tratamentos. No período de avaliação (maio a setembro/2001 - período seco), as disponibilidades de matéria seca total (DMST) e lâmina foliar (DMSLF) apresentaram valores proporcionais às doses de nitrogênio e potássio. A irrigação teve efeito significativo sobre a DMSLF. No entanto, os teores de FDN e FDA, ao longo do período seco, na área não-irrigada foram menores que na área irrigada. As concentrações de PB e dos macrominerais (P, K, Ca e Mg) não foram influenciadas pelas doses de N e K e pela irrigação, tanto na área irrigada como na não-irrigada.<br>An experiment was carried out in experimental plots of 300 m² in size to evaluate the effects of doses of NK fertilization and irrigation on forage mass yield and quality of elephantgras cv Napier during the dry season. The experimental design was completely randomized with three treatment replications. The experimental treatments were arranged in a 4 x 2 factorial combination of four N-K2O doses with and without irrigation, in the dry season. N-K2O doses were: 100-80; 200-160; 300-240 and 400-320 kg.ha-1, respectively. Figures for total dry forage mass (TDFM), and dry green leaf mass (DGLM) increased in proportion to N-K2O dose from May to September 2001, the dry season period. Irrigation had a marked significant effect on DGLM yield. Leaf samples from irrigated plots had higher contents of neutral and acid detergent fiber as compared to leaf samples from the control plots. Concentrations of crude protein, potash, phosphorus, calcium and magnesium did not respond either to N-K2O doses or to irrigation