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## ‣ Inequalities for zeros of associated polynomials and derivatives of orthogonal polynomials

Dimitrov, D. K.; Ronveaux, A.
Fonte: Elsevier B.V. Publicador: Elsevier B.V.
Tipo: Artigo de Revista Científica Formato: 321-331
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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.

## ‣ L-orthogonal polynomials associated with related measures

de Andrade, E. X. L.; Costa, M. S.; Ranga, Alagacone Sri
Fonte: Elsevier B.V. Publicador: Elsevier B.V.
Tipo: Artigo de Revista Científica Formato: 1041-1052
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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

## ‣ Szego type polynomials and para-orthogonal polynomials

Lamblem, R. L.; McCabe, J. H.; Pinar, M. A.; Ranga, Alagacone Sri
Tipo: Artigo de Revista Científica Formato: 30-41
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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.

## ‣ SZEGO and PARA-ORTHOGONAL POLYNOMIALS on THE REAL LINE: ZEROS and CANONICAL SPECTRAL TRANSFORMATIONS

Castillo, Kenier; Lamblem, Regina Litz; Rafaeli, Fernando Rodrigo; Ranga, Alagacone Sri
Fonte: Amer Mathematical Soc Publicador: Amer Mathematical Soc
Tipo: Artigo de Revista Científica Formato: 2229-2249
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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.

## ‣ Kernel polynomials from L-orthogonal polynomials

Felix, H. M.; Sri Ranga, A.; Veronese, D. O.
Tipo: Artigo de Revista Científica Formato: 651-665
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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

## ‣ Szego{double acute} and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations

Castillo, Kenier; LamblÉm, Regina Litz; Rafaeli, Fernando Rodrigo; Ranga, Alagacone Sri
Tipo: Artigo de Revista Científica Formato: 2229-2249
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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.

## ‣ Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle

Dimitrov, Dimitar K.; Ranga, A. Sri
Tipo: Artigo de Revista Científica
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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.

## ‣ Zeros of classical orthogonal polynomials of a discrete variable

Area, Ivan; Dimitrov, Dimitar K.; Godoy, Eduardo; Paschoa, Vanessa G.
Fonte: Amer Mathematical Soc Publicador: Amer Mathematical Soc
Tipo: Artigo de Revista Científica Formato: 1069-1095
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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.

## ‣ Zeros de polinômios ortogonais de variável discreta; Zeros of orthogonal polynomials of discrete variable

Vanessa Gonçalves Paschoa Ferraz
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
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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...

## ‣ Numerical and combinatorial applications of generalized Appell polynomials; Aplicações numéricas e combinatórias de polinómios de Appell generalizados

Cruz, Carla Maria
Português
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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...

## ‣ Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

Area, Iván; Godoy, Eduardo; Marcellán, Francisco; Moreno Balcázar, Juan José
Tipo: Artigo de Revista Científica Formato: application/pdf
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Let ${Q_n(x)}_n$ be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product $$igllangle (p(x),r(x)igr angle_S=igllangle{old u}_0,p(x)r(x) igr angle+ lambdaigllangle {old u}_1,(Delta p)(x)(Delta r)(x) igr angle,$$ 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...

## ‣ Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

Area, Iván; Godoy, Eduardo; Marcellán, Francisco; Moreno Balcázar, Juan José
Tipo: Artigo de Revista Científica Formato: application/pdf
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## ‣ Quantum algebras {SU}_q(2) and {SU}_q(1,1) associated with certain q-Hahn polynomials: a revisited approach

Arvesú, Jorge
Fonte: Kent State University Publicador: Kent State University
Tipo: Artigo de Revista Científica Formato: application/pdf
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...