J. Arvesú | Universidad Carlos III de Madrid (original) (raw)

Papers by J. Arvesú

Research paper thumbnail of Zeros of Jacobi and ultraspherical polynomials

Research paper thumbnail of Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Integral Transforms and Special Functions

Research paper thumbnail of Multiple Meixner Polynomials on a Non-Uniform Lattice

Mathematics

We consider two families of type II multiple orthogonal polynomials. Each family has orthogonalit... more We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.

Research paper thumbnail of On Infinitely Many Rational Approximants to ζ(3)

Mathematics

A set of second order holonomic difference equations was deduced from a set of simultaneous ratio... more A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

Research paper thumbnail of On two models of orthogonal polynomials and their applications

This contribution deals with some models of orthogonal polynomials as well as their applications ... more This contribution deals with some models of orthogonal polynomials as well as their applications in several areas of mathematics. Some new trends in the theory of orthogonal polynomials are summarized. In particular, we emphasize on two kinds of orthogonality, i.e., the standard orthogonality in the unit circle and a non standard one, which is called multi-orthogonality. Both have attracted the interest of researchers during the past ten years.

Research paper thumbnail of Recent trends in orthogonal polynomials and their applications

Research paper thumbnail of Special volume on orthogonal polynomials and approximation theory. Selected papers from the 5th international workshop on orthogonal polynomials, Madrid, Spain, June 24–27, 2002

Electronic transactions on numerical analysis ETNA

Research paper thumbnail of On new rational approximants to \zeta(3)

New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The ... more New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A comparison of our approximants with Ap\'ery's approximants to \zeta(3) is shown.

Research paper thumbnail of Casorati type determinants of some q-classical orthogonal polynomials

Proceedings of the American Mathematical Society, 2015

Some symmetries for Casorati determinants whose entries are q-classical orthogonal polynomials ar... more Some symmetries for Casorati determinants whose entries are q-classical orthogonal polynomials are studied. Special attention is paid to the symmetry involving Big q-Jacobi polynomials. Some limiting situations, for other related q-classical orthogonal polynomial families in the q-Askey scheme, namely q-Meixner, q-Charlier, and q-Laguerre polynomials, are considered.

Research paper thumbnail of On the q-Charlier Multiple Orthogonal Polynomials

Symmetry, Integrability and Geometry: Methods and Applications, 2015

We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials... more We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.

Research paper thumbnail of A Generalization of the Jacobi-Koornwinder Polynomials

Research paper thumbnail of On a modification of the Jacobi linear functional: Asymptotic properties and zeros of the corresponding orthogonal polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is rela... more 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 U U = J α,β + A 1 δ(x − 1) + B 1 δ(x + 1) − A 2 δ (x − 1) − B 2 δ (x + 1), where J α,β is the Jacobi linear functional, i.e. J α,β , p = 1 −1 p(x)(1 − x) α (1 + x) β dx, α, β > −1, p ∈ P, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1, 1) (inner asymptotics) and C \ [−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 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 ±1.

Research paper thumbnail of On the Connection and Linearization Problem for Discrete Hypergeometric q-Polynomials

Journal of Mathematical Analysis and Applications, 2001

In the present paper, starting from the second-order difference hypergeometric Ž. equation on the... more In the present paper, starting from the second-order difference hypergeometric Ž. equation on the non-uniform lattice x s satisfied by the set of discrete hypergeo-Ä 4 metric orthogonal q-polynomials p , we find analytical expressions of the expann Ž Ž .. Ž. sion coefficients of any q-polynomial r x s on x s and of the product m Ž Ž .. Ž Ž .. Ä 4 r x s q x s in series of the set p. These coefficients are given in terms of m j n the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials.

Research paper thumbnail of Asymptotics for multiple Meixner polynomials

Journal of Mathematical Analysis and Applications, 2014

We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respec... more We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively [7]. We use an algebraic function formulation for the solution of the equilibrium problem with constrain to describe their zero distribution. Then analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.

Research paper thumbnail of A high-order q -difference equation for q -Hahn multiple orthogonal polynomials

Journal of Difference Equations and Applications, 2012

Abstract A high-order linear q-difference equation with polynomial coefficients having q-Hahn mul... more Abstract A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some ...

Research paper thumbnail of First-order non-homogeneous q -difference equation for Stieltjes function characterizing q -orthogonal polynomials

Journal of Difference Equations and Applications, 2013

In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a... more In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn are worked out.

Research paper thumbnail of Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

Journal of Computational and Applied Mathematics, 1998

We obtain an explicit expression for the Sobolev-type orthogonal polynomials Q n (x) associated w... more We obtain an explicit expression for the Sobolev-type orthogonal polynomials Q n (x) associated with the inner product < p; q >= Z 1 ?1 p(x)q(x) (x)dx + A 1 p(1)q(1) + B 1 p(?1)q(?1) + A 2 p 0 (1)q 0 (1) + B 2 p 0 (?1)q 0 (?1); where (x) = (1?x) (1+x) is the Jacobi weight function, ; > ?1, A 1 ; B 1 ; A 2 ; B 2 0 and p, q 2 IP, the linear space of polynomials with real coe cients. The hypergeometric representation (6 F 5) and the second order linear di erential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in-1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q n (x). Such a zero is located outside the interval-1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.

Research paper thumbnail of Some discrete multiple orthogonal polynomials

Journal of Computational and Applied Mathematics, 2003

In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to p... more 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

Research paper thumbnail of On the Krall-type polynomials

Journal of Applied Mathematics, 2004

Using a general and simple algebraic approach, some results on Krall-type orthogonal polynomials ... more Using a general and simple algebraic approach, some results on Krall-type orthogonal polynomials and some of their extensions are obtained.

Research paper thumbnail of On the q -polynomials in the exponential lattice x ( s )= c 1 q s + c 3

Integral Transforms and Special Functions, 1999

Research paper thumbnail of Zeros of Jacobi and ultraspherical polynomials

Research paper thumbnail of Interlacing of zeros of Laguerre polynomials of equal and consecutive degree

Integral Transforms and Special Functions

Research paper thumbnail of Multiple Meixner Polynomials on a Non-Uniform Lattice

Mathematics

We consider two families of type II multiple orthogonal polynomials. Each family has orthogonalit... more We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.

Research paper thumbnail of On Infinitely Many Rational Approximants to ζ(3)

Mathematics

A set of second order holonomic difference equations was deduced from a set of simultaneous ratio... more A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

Research paper thumbnail of On two models of orthogonal polynomials and their applications

This contribution deals with some models of orthogonal polynomials as well as their applications ... more This contribution deals with some models of orthogonal polynomials as well as their applications in several areas of mathematics. Some new trends in the theory of orthogonal polynomials are summarized. In particular, we emphasize on two kinds of orthogonality, i.e., the standard orthogonality in the unit circle and a non standard one, which is called multi-orthogonality. Both have attracted the interest of researchers during the past ten years.

Research paper thumbnail of Recent trends in orthogonal polynomials and their applications

Research paper thumbnail of Special volume on orthogonal polynomials and approximation theory. Selected papers from the 5th international workshop on orthogonal polynomials, Madrid, Spain, June 24–27, 2002

Electronic transactions on numerical analysis ETNA

Research paper thumbnail of On new rational approximants to \zeta(3)

New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The ... more New (infinitely many) rational approximants to \zeta(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A comparison of our approximants with Ap\'ery's approximants to \zeta(3) is shown.

Research paper thumbnail of Casorati type determinants of some q-classical orthogonal polynomials

Proceedings of the American Mathematical Society, 2015

Some symmetries for Casorati determinants whose entries are q-classical orthogonal polynomials ar... more Some symmetries for Casorati determinants whose entries are q-classical orthogonal polynomials are studied. Special attention is paid to the symmetry involving Big q-Jacobi polynomials. Some limiting situations, for other related q-classical orthogonal polynomial families in the q-Askey scheme, namely q-Meixner, q-Charlier, and q-Laguerre polynomials, are considered.

Research paper thumbnail of On the q-Charlier Multiple Orthogonal Polynomials

Symmetry, Integrability and Geometry: Methods and Applications, 2015

We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials... more We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.

Research paper thumbnail of A Generalization of the Jacobi-Koornwinder Polynomials

Research paper thumbnail of On a modification of the Jacobi linear functional: Asymptotic properties and zeros of the corresponding orthogonal polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is rela... more 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 U U = J α,β + A 1 δ(x − 1) + B 1 δ(x + 1) − A 2 δ (x − 1) − B 2 δ (x + 1), where J α,β is the Jacobi linear functional, i.e. J α,β , p = 1 −1 p(x)(1 − x) α (1 + x) β dx, α, β > −1, p ∈ P, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1, 1) (inner asymptotics) and C \ [−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 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 ±1.

Research paper thumbnail of On the Connection and Linearization Problem for Discrete Hypergeometric q-Polynomials

Journal of Mathematical Analysis and Applications, 2001

In the present paper, starting from the second-order difference hypergeometric Ž. equation on the... more In the present paper, starting from the second-order difference hypergeometric Ž. equation on the non-uniform lattice x s satisfied by the set of discrete hypergeo-Ä 4 metric orthogonal q-polynomials p , we find analytical expressions of the expann Ž Ž .. Ž. sion coefficients of any q-polynomial r x s on x s and of the product m Ž Ž .. Ž Ž .. Ä 4 r x s q x s in series of the set p. These coefficients are given in terms of m j n the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials.

Research paper thumbnail of Asymptotics for multiple Meixner polynomials

Journal of Mathematical Analysis and Applications, 2014

We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respec... more We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively [7]. We use an algebraic function formulation for the solution of the equilibrium problem with constrain to describe their zero distribution. Then analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.

Research paper thumbnail of A high-order q -difference equation for q -Hahn multiple orthogonal polynomials

Journal of Difference Equations and Applications, 2012

Abstract A high-order linear q-difference equation with polynomial coefficients having q-Hahn mul... more Abstract A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some ...

Research paper thumbnail of First-order non-homogeneous q -difference equation for Stieltjes function characterizing q -orthogonal polynomials

Journal of Difference Equations and Applications, 2013

In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a... more In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn are worked out.

Research paper thumbnail of Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

Journal of Computational and Applied Mathematics, 1998

We obtain an explicit expression for the Sobolev-type orthogonal polynomials Q n (x) associated w... more We obtain an explicit expression for the Sobolev-type orthogonal polynomials Q n (x) associated with the inner product < p; q >= Z 1 ?1 p(x)q(x) (x)dx + A 1 p(1)q(1) + B 1 p(?1)q(?1) + A 2 p 0 (1)q 0 (1) + B 2 p 0 (?1)q 0 (?1); where (x) = (1?x) (1+x) is the Jacobi weight function, ; > ?1, A 1 ; B 1 ; A 2 ; B 2 0 and p, q 2 IP, the linear space of polynomials with real coe cients. The hypergeometric representation (6 F 5) and the second order linear di erential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in-1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q n (x). Such a zero is located outside the interval-1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.

Research paper thumbnail of Some discrete multiple orthogonal polynomials

Journal of Computational and Applied Mathematics, 2003

In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to p... more 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

Research paper thumbnail of On the Krall-type polynomials

Journal of Applied Mathematics, 2004

Using a general and simple algebraic approach, some results on Krall-type orthogonal polynomials ... more Using a general and simple algebraic approach, some results on Krall-type orthogonal polynomials and some of their extensions are obtained.

Research paper thumbnail of On the q -polynomials in the exponential lattice x ( s )= c 1 q s + c 3

Integral Transforms and Special Functions, 1999