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Juan L. Varona

Publications and source records attributed to Juan L. Varona.

10 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↗
arXiv2023

The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series

We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $ζ(2k+1)$ and $β(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-μ)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-μ)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials.

math.NT↗
arXiv2023

Three essays on Machin's type formulas

We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$. These formulas are computationally useful because multiplication or division by a power of two is a very fast operation for most computers. The second one presents a method for finding infinitely many formulas with $N$ terms. In the particular case $N=2$ the method is quite useful. It recovers most known formulas, gives some new ones, and allows to prove in an easy way that there are two terms Machin formulas with Lehmer measure as small as desired. Finally, we correct an oversight from previous result and give all Machin's type formulas with two terms involving arctangents of powers of the golden section.

math.NT↗
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

A Couple of Transcendental Prime-Representing Constants

It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be proved, but it is not known whether these constants are algebraic or transcendental numbers. By using Liouville or Roth's theorems about approximation by rationals, we find a couple of prime-representing constants that can be proved to be transcendental numbers.

math.NT↗
arXiv2015

Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in $L_p$ does not converge unless $p=2$. As a by-product of our work on quasi-greedy bases in $L_{p}(μ)$, we show that no normalized unconditional basis in $L_p$, $p\not=2$, can be semi-normalized in $L_q$ for $q\not=p$, thus extending a classical theorem of Kadets and Pełczy{ń}ski from 1968.

math.FA↗
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↗
arXiv2011

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;λ)$ in detail. The starting point is their Fourier series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials $\mathcal{E}_{n}(x;λ)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.

math.NT↗
arXiv2002

Singular measures and convolution operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1,1) inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension.

math.CA↗