arXiv subjects

Mario Pérez

Publications and source records attributed to Mario Pérez.

6 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↗

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↗

arXiv2016

Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials with applications to series involving zeros of Bessel functions

We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a bunch of series involving the zeros of Bessel functions.

math.CA↗

arXiv2014

Admissibility condition for exceptional Laguerre polynomials

We prove a necessary and sufficient condition for the integrability of the weight associated to the exceptional Laguerre polynomials. This condition is very much related to the fact that the associated second order differential operator has no singularities in $(0,+\infty)$.

math.CA↗

arXiv2013

Misfortunes of a mathematicians' trio using Computer Algebra Systems: Can we trust?

Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of matrices of integer numbers: not only it computes the determinants wrongly, but also it produces different results if one evaluates the same determinant twice.

cs.SC↗

arXiv1995

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $ν$ on $[-1,1]$ are studied when $ν$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by $b \in \BMO$. We also prove some norm inequalities for $S_n$ when $ν$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$.

math.CA↗