It is well known and easy to see that the zeros of both the associated polynomial and the derivative of an orthogonal polynomial p(n)(x) interlace with the zeros of p(n)(x) itself. The natural question of how these zeros interlace is under discussion. We give a sufficient condition for the mutual location of kth, 1 less than or equal to k less than or equal to n - 1, zeros of the associated polynomial and the derivative of an orthogonal polynomial in terms of inequalities for the corresponding Cotes numbers. Applications to the zeros of the associated polynomials and the derivatives of the classical orthogonal polynomials are provided. Various inequalities for zeros of higher order associated polynomials and higher order derivatives of orthogonal polynomials are proved. The results involve both classical and discrete orthogonal polynomials, where, in the discrete case, the differential operator is substituted by the difference operator. (C) 2001 IMACS. Published by Elsevier B.V. B.V. All rights reserved.

A positive measure psi defined on [a, b] such that its moments mu(n) = integral(b)(a)t(n) d psi(t) exist for n = 0, +/-1, +/-2. can be called a strong positive measure on [a, b] When 0 <= a < b <= infinity the sequence of polynomials {Q(n)} defined by integral(b)(a) t(-n+s) Q(n)(t) d psi(t) = 0, s = 0, ., n - 1, exist and they are referred here as L-orthogonal polynomials We look at the connection between two sequences of L-orthogonal polynomials {Q(n)((1))} and {Q(n)((0))} associated with two closely related strong positive measures and th defined on [a, b]. To be precise, the measures are related to each other by (t - kappa) d psi(1)(t) = gamma d psi(0)(t). where (t - kappa)/gamma is positive when t is an element of (n, 6). As applications of our study. numerical generation of new L-orthogonal polynomials and monotonicity properties of the zeros of a certain class of L-orthogonal polynomials are looked at. (C) 2010 IMACS Published by Elsevier B V All rights reserved

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Szego type polynomials with respect to a linear functional M for which the moments M[t(n)] = mu(-n) are all complex, mu(-n) = mu(n) and D(n) not equal 0 for n >= 0. are considered. Here, D(n) are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szego type polynomials are also given. Examples are provided to justify the results. (C) 2010 Elsevier B.V. All rights reserved.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Processo FAPESP: 09/13832-9; We study polynomials which satisfy the same recurrence relation as the Szego polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szego polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szego polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szego polynomials and polynomials arising from canonical spectral transformations are obtained.

A positive measure ψ defined on [a,b] such that its moments μn=∫a btndψ(t) exist for n=0,±1,±2,⋯, is called a strong positive measure on [a,b]. If 0≤a

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We study polynomials which satisfy the same recurrence relation as the Szego{double acute} polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szego{double acute} polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szego{double acute} polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szego{double acute} polynomials and polynomials arising from canonical spectral transformations are obtained. © 2012 American Mathematical Society.

Para-orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para-orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para-orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner-Pollaczek polynomials is proved. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Processo FAPESP: 09/13832-9; In this paper we obtain sharp bounds for the zeros of classical orthogonal polynomials of a discrete variable, considered as functions of a parameter, by using a theorem of A. Markov and the so-called Hellmann-Feynman theorem. Comparisons with previous results for zeros of Hahn, Meixner, Kravchuk and Charlier polynomials are also presented.

Neste trabalho estudamos o comportamento de zeros de polinômios ortogonais clássicos de variável discreta. Provamos que certas funções que envolvem os zeros dos polinômios de Charlier, Meixner, Kravchuck e Hahn são funções monótonas dos parâmetros dos quais os correspondentes polinômios dependem. Com esse resultado obtemos novos limitantes extremamente precisos para os zeros dessas famílias de polinômios em função dos zeros dos polinômios ortogonais clássicos, que são mais estudados. Analisamos quais são os melhores limitantes explícitos para os zeros desses polinômios e aplicamos aos nossos resultados, obtendo assim limitantes explícitos para os zeros dos polinômios de Charlier, Meixner, Kravchuck e Hahn. São feitas comparações entre os nossos resultados e os melhores resultados encontrados na literatura para os zeros desses polinômios e verifica-se que nossos limitantes são, em uma grande parte, melhores. Devido à sua grande aplicabilidade, um estudo ainda mais minucioso foi feito para os zeros dos polinômios de Gram, um caso particular de Hahn, que resultou em limitantes para os zeros dos polinômios de Gram. Experimentos numéricos comprovam a qualidade dos resultados.; In this thesis we study the behavior of zeros of classical orthogonal polynomials of discrete variable. We prove that certain functions which involve the zeros of polynomials of Charlier...

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.; Esta tese estuda propriedades e aplicações de diferentes polinómios de Appell generalizados no contexto da análise de Clifford. Exemplificando uma transformação realizada por polinómios de Appell generalizados...

Let ${Q_n(x)}_n$ be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product $$igllangle (p(x),r(x)igr angle_S=igllangle{old u}_0,p(x)r(x) igr angle+ lambdaigllangle {old u}_1,(Delta p)(x)(Delta r)(x) igr angle,$$ where $lambdage 0$, $(Delta f)(x)=f(x+1)-f(x)$ denotes the forward difference operator and $({old u}_0,{old u}_1)$ is a $Delta$-coherent pair of positive-definite linear functionals being ${old u}_1$ the Meixner linear functional. In this paper, relative asymptotics for the ${Q_n(x)}_n$ sequence with respect to Meixner polynomials on compact subsets of $bfCsetminus[0,+infty)$ is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-$Delta$-coherent pair, that is, when ${old u}_0={old u}_1$ is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.; The work by I.A. and E.G. was partially supported by Ministerio de Ciencia y Tecnología of Spain under grant BFM2002-04314-C02-01. The work by F.M. has been supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2003-06335-C03-02 as well as by the NATO collaborative grant PST.CLG. 979738. The work by J.J.M.B has been supported by Dirección General de Investigación of Spain under grant BFM2001-3878-C02-02 as well as by Junta de Andalucía (research group FQM0229).; 16 pages...

We study the analytic properties of the monic Meixner-Sobolev polynomials ${Q_n}$ orthogonal with respect to the inner product involving differences $$(p,q)_S=sum infty_{i=0}[p(i)q(i)+lambdaDelta p(i)Delta q(i)] {mu (gamma)_iover i!},$$ $gamma>0, 0

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In this paper, polynomials which are orthogonal with respect to the inner product $$langle p,q angle_S= sum infty_{s=0} p(s)q(s) {mu Gamma (gamma+s) overGamma(s+1) Gamma (gamma)}+ lambda sum infty_{s=0} Delta p(s)Delta q(s){mu Gamma(gamma+s) overGamma (s+1)Gamma (gamma)},$$ where $00$, $lambdage 0$ are studied. For these polynomials, algebraic properties and difference equations are obtained as well as their relation with the Meixner polynomials. Moreover, some properties about the zeros of these polynomials are deduced.; The work of I.A. and E.G. has been partially supported by Xunta de Galicia-Universidade de Vigo under grant 64502I703. E.G. also wishes to acknowledge partial support by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB-95-0828. The research of F.M. was partially supported by DGES of Spain under Grant PB96-1020-C03-01 and INTAS Project 93-0219 Ext.; 31 pages, no figures.-- MSC2000 codes: 33C45, 33D45, 39A10, 39A70, 42C05.; MR#: MR1752153 (2000m:33006); Zbl#: Zbl 0948.33004

In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317–347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally...

The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional L, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with two new examples of linear spectral transformations. First, we analyze a spectral transformation of L such that the corresponding moment matrix is the result of the addition of a constant on the main diagonal or on two symmetric sub-diagonals of the initial moment matrix. Next, we introduce a spectral transformation of L by the addition of the first derivative of a complex Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and su cient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. We also consider polynomials which satisfy the same recurrence relation as the polynomials orthogonal with respect to the linear functional L ...

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Processo FAPESP: 2009/13832-9; Processo FAPESP: 2013/23606-1; Sharp bounds for the zeros of symmetric Kravchuk polynomials K (n) (x;M) are obtained. The results provide a precise quantitative meaning of the fact that Kravchuk polynomials converge uniformly to Hermite polynomials, as M tends to infinity. They show also how close the corresponding zeros of two polynomials from these sequences of classical orthogonal polynomials are.

In the last 30 years, the study of orthogonal polynomials in Sobolev spaces has obtained an increasing attention from the research community. The first work on Sobolev orthogonal polynomials was published in 1962 by Althammer, who studied the Legendre-Sobolev polynomials orthogonal with respect to the inner product. The study of this family of orthogonal polynomials is not only interesting for a comparison with the standard theory of orthogonal polynomials, but these polynomials also arise in a natural way in a variety of contexts. In this thesis, we analyze the properties of polynomials orthogonal with respect to a discrete Sobolev inner product. More precisely, we will focus our attention on the study of connection formulas relating Sobolev orthogonal polynomials with the corresponding ordinary ones. Indeed, we deal with some problems on asymptotic behavior of Sobolev orthogonal polynomials as well as we obtain some results on convergence of Fourier-Sobolev series. The present Thesis is organized as follows: In Chapter 1 we introduce the theory of Sobolev orthogonal polynomials and the notation that we will use along this Thesis. We summarize two main differences between the standard orthogonal polynomials and the Sobolev case: recurrence relations and the location of zeros of orthogonal polynomials. Here...

We study properties of the monic polynomials $\{Q_n\}_{n\in\bbfN}$ orthogonal with respect to the Sobolev inner product $$(p,q)_S= \int infty_0 (p,p')\pmatrix 1 & \mu\\ \mu &\lambda\endpmatrix \pmatrix q\\ q'\endpmatrix x alpha e -x} dx,$$ where $\lambda- \mu > 0$ and $\alpha> -1$. This inner product can be expressed as $$(p,q)_S= \int infty_0 p(x) q(x)((\mu+ 1) x- \alpha\mu) x \alpha- 1} e -x} dx+ \lambda\int infty_0 p'q' x alpha e -x} dx,$$ when $\alpha> 0$. In this way, the measure which appears in the first integral is not positive on $[0,\infty)$ for $\mu\in \bbfR\setminus[- 1,0]$. The aim of this paper is the study of analytic properties of the polynomials $Q_n$. First, we give an explicit representation for $Q_n$ using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for $\widetilde k_n= (Q_n,Q_n)_S$. Then we consider analytic aspects. We first establish the strong asymptotics of $Q_n$ on $\bbfC\setminus[0,\infty)$ when $\mu\in \bbfR$ and we also obtain an asymptotic expression on the oscillatory region, that is, on $(0,\infty)$. Then we study the Plancherel-Rotach asymptotics for the Sobolev polynomials $Q_n(nx)$ on $\bbfC\setminus[0, 4]$ when $\mu\in (- 1,0]$. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of $Q_n$. We also give a Mehler-Heine type formula for the Sobolev polynomials which is valid on compact subsets of $\bbfC$ when $\mu\in (-1...

This contribution deals with the connection of q-Clebsch-Gordan coefficients $(q$-CGC) of the Wigner-Racah algebra for the quantum groups $SU_q(2)$ and $SU_q(1,1)$ with certain q-Hahn polynomials. A comparative analysis of the properties of these polynomials and $su_q (2)$ and $su_q(1,1)$ Clebsch-Gordan coefficient shows that each relation for q-Hahn polynomials has the corresponding partner among the properties of q-CGC and vice versa. Consequently, special emphasis is given to the calculations carried out in the linear space of polynomials, i.e., to the main characteristics and properties for the new q-Hahn polynomials obtained here by using the Nikiforov-Uvarov approach [{\it A. F. Nikiforov}, {\it S. K. Suslov} and {\it V. B. Uvarov}, Orthogonal Polynomials in Discrete Variables, Springer-Verlag, Berlin, 1991; {\it A. F. Nikiforov} and {\it V. B. Uvarov}, Classical orthogonal polynomials in a discrete variable on non-uniform lattices, Preprint Inst. Prikl. Mat. M. V. Keldysh Akad. Nauk SSSR (In Russian), 17, Moscow, 1983] on the non-uniform lattice $x(s)=\frac{q -1}{q-1}$. These characteristics and properties will be important to extend the q-Hahn polynomials to the multiple case [{\it J. Arvesú}, q-Discrete multiple orthogonal polynomials...

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional ${\scr U}$ $$ {\scr U}={\scr J}_{\alpha,\beta}+A_1\delta(x-1)+B_1\delta(x+1)- A_2\delta'(x-1)-B_2\delta'(x+1), $$ where ${\scr J}_{\alpha,\beta}$ is the Jacobi linear functional, i.e. $$ \langle{\scr J}_{\alpha,\beta},p\rangle=\int ^1_{-1}p(x)(1-x)^\alpha (1+x)^\beta dx,\quad \alpha,\beta>-1,\ p\in {\Bbb P}, $$ and ${\Bbb P}$ is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in $(-1,1)$ (inner asymptotics) and ${\Bbb C}\sbs [-1,1]$ (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional ${\scr U}$ is a generalization of one studied by T. H. Koornwinder when $A_2=B_2=0$. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi-Markov function by a rational function with two double poles at $\pm 1$. The denominators of the $[n-1/n]$ Padé approximants are our orthogonal polynomials.; The first author (J.A.) was partially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM2000-0029 and BFM2000-0206-C04-01. The research of the authors (F.M. and R.A.N.) was partially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM2000-0206-C04-01 and BFM2000-0206-C04-02...