J. Varona

activity

19952026

most citedHarmonic Analysis associated with a discrete Laplacian

85 citations · 282 across the 23 of their papers we have counts for

collaborators

25 papers

math.GM2026

Zeros of GKP sequences of polynomials

Antonio J. Durán, Mario Pérez, Juan L. Varona

Given two sequences $ϕ = (ϕ_{i})_{i \geq 1}$ and $ψ = (ψ_{i})_{i \geq 1}$ and numbers $a, b, c$ , we introduce the GKP sequence of polynomials $(p_{n})_{n}$ using the following recurrence formula: $p…

math.CO2025★ 2 cited

A class of weighted Delannoy numbers

J. M. Grau, A. M Oller-Marcen, J. L. Varona

The weighted Delannoy numbers are defined by the recurrence relation $f_{m, n} = α f_{m - 1, n} + β f_{m, n - 1} + γ f_{m - 1, n - 1}$ if $mn > 0$ , with $f_{m, n} = α^{m} β^{n}$ if $nm = 0$ . In this work, we…

math.NT2023

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

Óscar Ciaurri, Luis M. Navas, Francisco J. Ruiz +1

We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] w…

math.NT2023★ 7 cited

Three essays on Machin's type formulas

Armengol Gasull, Florian Luca, Juan L. Varona

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 va…

math.CA2022★ 2 cited

Summing Sneddon-Bessel series explicitly

Antonio J. Durán, Mario Pérez, Juan L. Varona

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 $0 < x$ , $0 < y$ , $x + y < 2$ …

math.NT2021

On the Solution of the Equation $n = ak + b p_{k}$ by Means of an Iterative Method

Juan Luis Varona

For fixed positive integers $n$ , we study the solution of the equation $n = k + p_{k}$ , where $p_{k}$ denotes the $k$ th prime number, by means of the iterative method \[ k_{j+1} = π(n-…

math.NT2020

A Couple of Transcendental Prime-Representing Constants

Juan L. Varona

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…

math.NT2020

Continued fractions for strong Engel series and Lüroth series with signs

Andrew N. W. Hone, Juan Luis Varona

An Engel series is a sum of reciprocals $\sum_{j \geq 1} 1/ x_{j}$ of a non-decreasing sequence of positive integers $x_{n}$ with the property that $x_{n}$ divides $x_{n + 1}$ for all $n\geq…

math.NT2018★ 10 cited

Continued fractions and irrationality exponents for modified Engel and Pierce series

Andrew N. W. Hone, Juan Luis Varona

An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_{n})$ of positive integers, which is such that each term is divisible by the previous one, and a Pierce serie…

math.CA2016

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

Óscar Ciaurri, Antonio J. Durán, Mario Pérez +1

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 th…

math.AP2016★ 16 cited

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

Ó. Ciaurri, L. Roncal, P. R. Stinga +2

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h > 0$ \[ (-Δ_h)^su=f, \] for $u, f : Z_{h} \to R$ , $0…

math.NA2016

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

Gunar Matthies, Mehdi Salimi, Somayeh Sharifi +1

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function ev…

math.NA2015★ 1 cited

An optimal class of eighth-order iterative methods based on Kung and Traub's method with its dynamics

Gunar Matthies, Mehdi Salimi, Somayeh Sharifi +1

In this paper, we present a three-point without memory iterative method based on Kung and Traub's method for solving non-linear equations in one variable. The proposed method has e…

math.FA2015

Unconditional and quasi-greedy bases in $L_{p}$ with applications to Jacobi polynomials Fourier series

Fernando Albiac, José L. Ansorena, Óscar Ciaurri +1

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…

math.AP2015★ 11 cited

Fractional discrete Laplacian versus discretized fractional Laplacian

Ó. Ciaurri, L. Roncal, P. R. Stinga +2

We define and study some properties of the fractional powers of the discrete Laplacian $(- Δ_{h})^{s}, on Z_{h} = h Z,$ for $h > 0$ and $0 < s < 1$ . A comparison b…

math.CA2014★ 85 cited

Harmonic Analysis associated with a discrete Laplacian

Ó. Ciaurri, T. A. Gillespie, L. Roncal +2

It is well-known that the fundamental solution of $u_{t} (n, t) = u (n + 1, t) - 2 u (n, t) + u (n - 1, t), n \in Z,$ with $u (n, 0) = δ_{nm}$ for every fixed $m \in Z$ , is given…

cs.SC2013★ 50 cited

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

Antonio J. Durán, Mario Pérez, Juan L. Varona

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 det…

math-ph2012★ 3 cited

A $q$ -linear analogue of the plane wave expansion

Luís Daniel Abreu, Óscar Ciaurri, Juan Luis Varona

We obtain a $q$ -linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$ -Gegenbauer polynomials and the \textit{third} Jackson $q$ -Bes…

math.NT2012★ 16 cited

A simple computation of $ζ (2 k)$ by using Bernoulli polynomials and a telescoping series

Ó. Ciaurri, L. M. Navas, F. J. Ruiz +1

We present a new proof of Euler's formulas for $ζ (2 k)$ , where $k = 1, 2, 3, ...$ , which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $ζ (2 k)$ …

math.NT2011★ 37 cited

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

Luis M. Navas, Francisco J. Ruiz, Juan L. Varona

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $B_{n} (x; λ)$ in detail. The starting point is their Fourier series on $[0, 1]$ which, it is shown,…

math.FA2010★ 10 cited

Mean convergence of Fourier-Dunkl series

Ó. Ciaurri, M. Pérez, J. M. Reyes +1

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion as…

math.FA2009★ 14 cited

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

L. D. Abreu, Ó. Ciaurri, J. L. Varona

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structu…

math.FA2005★ 3 cited

Heat and Poisson semigroups for Fourier-Neumann expansions

J. J. Betancor, O. Ciaurri, T. Martínez +3

Given $α > - 1$ , consider the second order differential operator in $(0, \infty)$ , $L_{α} f \equiv (x^{2} \frac{d ^{2}}{d x ^{2}} + (2 α + 3) x \frac{d}{d x} + x^{2} + (α + 1)^{2}) (f),$ which appears in…

math.CA2002★ 3 cited

Singular measures and convolution operators

J. M. Aldaz, Juan L. Varona

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) inequaliti…

Zeros of GKP sequences of polynomials

A class of weighted Delannoy numbers

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

Three essays on Machin's type formulas

Summing Sneddon-Bessel series explicitly

On the Solution of the Equation n=ak+bpk​ by Means of an Iterative Method

A Couple of Transcendental Prime-Representing Constants

Continued fractions for strong Engel series and Lüroth series with signs

Continued fractions and irrationality exponents for modified Engel and Pierce series

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

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

An optimal class of eighth-order iterative methods based on Kung and Traub's method with its dynamics

Unconditional and quasi-greedy bases in Lp​ with applications to Jacobi polynomials Fourier series

Fractional discrete Laplacian versus discretized fractional Laplacian

Harmonic Analysis associated with a discrete Laplacian

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

A q-linear analogue of the plane wave expansion

A simple computation of ζ(2k) by using Bernoulli polynomials and a telescoping series

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

Mean convergence of Fourier-Dunkl series

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

Heat and Poisson semigroups for Fourier-Neumann expansions

Singular measures and convolution operators

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

On the Solution of the Equation $n = ak + b p_{k}$ by Means of an Iterative Method

Unconditional and quasi-greedy bases in $L_{p}$ with applications to Jacobi polynomials Fourier series

A $q$ -linear analogue of the plane wave expansion

A simple computation of $ζ (2 k)$ by using Bernoulli polynomials and a telescoping series