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Antonio J. Durán

Publications and source records attributed to Antonio J. Durán.

At least 19 recordsLinked to original sources
arXiv2026

Zeros of GKP sequences of polynomials

Given two sequences $ϕ=(ϕ_i)_{i\ge 1}$ and $ψ=(ψ_i)_{i\ge 1}$ and numbers $a,b,c$, we introduce the GKP sequence of polynomials $(p_n)_n$ using the following recurrence formula: $p_0 = 1$ and for $n\ge 1$ \[ p_{n}(x) = (ax^2+bx+c) p_{n-1}'(x) + (ϕ_{n} + ψ_{n} x)p_{n-1}(x), \] where we assume that $ax^2+bx+c$ has two different real zeros. Tangent, Secant, Eulerian or Jacobi polynomials are examples of GKP sequences of polynomials. In this paper, under mild assumptions we prove that the zeros of the polynomials $p_n$ are real, simple and live between the zeros of $ax^2+bx+c$. Moreover, the zeros of $p_{n+1}$ interlace the zeros of $p_n$. We study in detail the cases when $ψ$ is constant, and $ϕ=(ϕ_i)_{i\ge 1}$ is constant for $i$ big enough, proving, among other results, asymptotics for the leftmost and rightmost zeros of $p_n$.

math.GM↗
arXiv2025

Using quasi-Darboux transformations to construct exceptional matrix polynomials

We introduce a couple of methods to construct exceptional matrix polynomials. One of them uses what we have called quasi-Darboux transformations. This seems to be a more powerful method to deal with the non-commutativity problems that appear when matrix-valued polynomials are considered. The other method does not use any transformation of Darboux type. Using both methods, we construct a collection of five illustrative examples that show how powerful our two methods are. The examples include exceptional matrix polynomials of Hermite, Laguerre, and Gegenbauer type, as well as an example with a weight matrix having a Dirac delta.

math.CA↗
arXiv2025

Zeros of linear combinations of Laguerre polynomials

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_j\tilde L^α_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^α$, the polynomials $\mathcal L_n^α=n!L_n^α/(1+α)_n$ (so that $\mathcal L_n^α(0)=1$), the standard Laguerre polynomials $(L_n^α)_n$ and the Brenke normalization $L_n^α/(1+α)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^jγ_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^Kγ_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $α+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough.

math.CA↗
arXiv2025

Zeros of linear combinations of Hermite polynomials

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_j\tilde H_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\dots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$. We consider two different normalizations of Hermite polynomials: the standard one (i.e. $\tilde H_n=H_n$), and $\tilde H_n=H_n/(2^nn!)$ (so that $q_n$ are Appell polynomials: $q_n'=q_{n-1}$). In both cases, we show the key role played by the polynomial $P(x)=\sum_{j=0}^Kγ_jx^{K-j}$ to solve this problem. In particular, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real.

math.CA↗
arXiv2025

Zeros of linear combinations of orthogonal polynomials

Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_jp_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $μ$ there always exists a sequence of orthogonal polynomials with respect to $μ$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $γ_j$'s.

math.CA↗
arXiv2025

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form $$ W(t)=t^αe^{-t}e^{At}t^{B}t^{B^*}e^{A^* t}, $$ where $A$ and $B$ are certain (nilpotent and diagonal, respectively) $N\times N$ matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.

math.CA↗
arXiv2024

Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

Write $ζ_m(n)$, $1\le m\le n-1$, for the negative zeros of the $n$-th Bell polynomial, ordered in decreasing size. In this paper, we prove the following asymptotic: for a positive integer $m$ we have $$ \lim_{n\to \infty}\frac{ζ_m(n)}{-m\left(\displaystyle\frac{m}{m+1}\right)^{n-1}}=1. $$ The approach used to find this asymptotic applies to many other significant families of polynomials. In particular, analogous asymptotics are also proved for the negative rightmost zeros of Eulerian polynomials, $r$-Bell polynomials, linear combinations of $K$ consecutive Bell polynomials and many others.

math.NT↗
arXiv2024

Generalized Bell polynomials

In this paper, generalized Bell polynomials $(\Be_n^ϕ)_n$ associated to a sequence of real numbers $ϕ=(ϕ_i)_{i=1}^\infty$ are introduced. Bell polynomials correspond to $ϕ_i=0$, $i\ge 1$. We prove that when $ϕ_i\ge 0$, $i\ge 1$: (a) the zeros of the generalized Bell polynomial $\Be_n^ϕ$ are simple, real and non positive; (b) the zeros of $\Be_{n+1}^ϕ$ interlace the zeros of $\Be_n^ϕ$; (c) the zeros are decreasing functions of the parameters $ϕ_i$. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.

math.CA↗
arXiv2024

Brenke polynomials with real zeros and the Riemann Hypothesis

If $A(z)=\sum_{n=0}^\infty a_nz^n$ and $B(z)=\sum_{n=0}^\infty b_nz^n$ are two formal power series, with $a_n,b_n\in \mathbb{R}$, the polynomials $(p_n)_n$ defined by the generating function $$ A(z)B(xz)=\sum_{n=0}^\infty p_n(x)z^n $$ are called the Brenke polynomials generated by $A$ and associated to $B$. We say that $A\in \mathcal{R}_B$ if the Brenke polynomials $(p_n)_n$ have only real zeros. Among other results, in this paper we find necessary and sufficient conditions on $B$ such that $\mathcal{R}_B=\mathcal{L}\text{-}\mathcal{P}$, where $\mathcal{L}\text{-}\mathcal{P}$ denotes the Laguerre-Pólya class (of entire functions). These results can be considered an extension to Brenke polynomials of the Jensen, and Pólya and Schur characterization $\mathcal{R}_{e^z}=\mathcal{L}\text{-}\mathcal{P}$, for Appell polynomials. When applying our results to a relative of the Riemann zeta function, we find new equivalencies for the Riemann Hypothesis in terms of real-rootedness of some sequences of Brenke polynomials.

math.CA↗
arXiv2022

Summing Sneddon-Bessel series explicitly

We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty \frac{J_α(x j_{m,ν})J_β(y j_{m,ν})} {j_{m,ν}^{2n+α+β-2ν+2} J_{ν+1}(j_{m,ν})^2}, \] where $0<x$, $0<y$, $x+y<2$, $n$ is an integer, $α,β,ν\in \mathbb{C}\setminus \{-1,-2,\dots \}$ with $2\operatorname{Re} ν< 2n+1 + \operatorname{Re} α+ \operatorname{Re} β$ and $\{j_{m,ν}\}_{m\geq 0}$ are the zeros of the Bessel function $J_ν$ of order $ν$. As an application we prove some extensions of the Kneser-Sommerfeld expansion.

math.CA↗
arXiv2021

New examples of Krall-Meixner and Krall-Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials

We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $ρ_{a,c}$ and Hahn measure $ρ_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre polynomials.

math.CA↗
arXiv2021

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second order difference or differential operator. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

math.CA↗
arXiv2020

Bispectral Jacobi type polynomials

We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results we prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line.

math.CA↗
arXiv2019

Bispectral Laguerre type polynomials

We study the bispectrality of Laguerre type polynomials, which are defined by taking suitable linear combinations of a fixed number of consecutive Laguerre polynomials. These Laguerre type polynomials are eigenfunctions of higher-order differential operators and include, as particular cases, the Krall-Laguerre polynomials. As the main results, we prove that these Laguerre type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Laguerre families are the only polynomials which are orthogonal with respect to a measure on the real line.

math.CA↗
arXiv2018

On difference operators for symmetric Krall-Hahn polynomials

The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete measures by suitable polynomials. This problem was raised by Richard Askey in 1991. The case of the Hahn family is specially rich due to the number of parameters involved. These suitable polynomials are in the Hahn case generated from a quartet $F_1,F_2,F_3$ and $F_4$ of finite sets of positive integers. In this paper, we show that the order of the corresponding difference operators can be reduced by assuming certain symmetries on both, the parameters of the Krall-Hahn family and the associated quartet $F_1,F_2,F_3$ and $F_4$.

math.CA↗
arXiv2016

From Krall discrete orthogonal polynomials to Krall polynomials

We show how to get Krall polynomials from Krall discrete polynomials using a procedure of passing to the limit in some of the parameters of the family. We also show that this procedure has to be different to the standard one used in the Askey scheme to go from the classical discrete polynomials to the classical polynomials.

math.CA↗
arXiv2015

Differential equations for discrete Jacobi-Sobolev orthogonal polynomials

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at $-1$ and/or $+1$. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters $α$ and $β$ are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrices which define the discrete Jacobi-Sobolev bilinear form.

math.CA↗