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Jesús Guillera

Publications and source records attributed to Jesús Guillera.

At least 19 recordsLinked to original sources
arXiv2025

Fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$

We prove two fast formulas for the Hurwitz values $ζ(2,a)$ and $ζ(3,a)$ respectively with the help of the WZ method. In them $(a)_n$ denotes the rising factorial or Pochhammer's symbol defined by $(a)_0=1$ and $(a)_n=a(a+1)\cdots(a+n-1)$ for positive integers $n$. The Huwitz $ζ$ function is defined by $ζ(s,a)=ζ(0,s,a)=\sum_{k=0}^{\infty} (k+a)^{-s}$. In addition, we can use these fast evaluations to compute also in a rapid way Dirichlet values of the kinds $L_χ(2)$ and $L_χ(3)$.

math.NT↗
arXiv2025

The fastest series for $1/π$ due to Ramanujan. Proofs from modular polynomials

First we give general formulas for proving real or complex Ramanujan series for $1/π$. Then, as an example, we apply them for providing complete proofs of the fastest series for $1/π$ due to Ramanujan using Russell and Weber modular polynomials. We recommend the reader to use a Maple program which is in the web of the author for automatically proving any Ramanujan-type series for $1/π$.

math.NT↗
arXiv2025

The WZ method and flawless WZ pairs

Recently, Kam Cheong Au discovered a powerful methodology of finding new Wilf-Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of $π$ and their duals for Dirichlet $L$-values. In this note we explain how a modification of Au's WZ pairs together with a classical analytic argument allows one to obtain simpler proofs of his results. We illustrate our method with a few examples elaborated with assistance of Maple code that we have developed.

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arXiv2020

Proof of Chudnovskys' series for $1/π$

We prove rational alternating Ramanujan-type series of level $1$ discovered by the brothers David and Gregory Chudnovky, by using a method of the author. We have carried out the computations with Maple (a symbolic software for mathematics).

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arXiv2019

Bilateral Ramanujan-like series for $1/π^k$ and their congruences

We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $π$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In addition we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation.

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arXiv2018

A method for proving Ramanujan series for $1/π$

In a famous paper of $1914$ Ramanujan gave a list of $17$ extraordinary formulas for the number $π$. In this paper we explain a general method to prove them, based on an original idea of James Wan and in some own ideas.

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arXiv2018

Ramanujan series with a shift

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_0$, we call the resulting series: "Ramanujan series with the shift $x_0$". Then, we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_0=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell=1, 2, 3$ and the shift $x_0=1/2$.

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arXiv2017

A family of Ramanujan-Orr formulas for $1/π$

We use a variant of Wan's method to prove two Ramanujan-Orr type formulas for $1/π$. This variant needs to know in advance the formulas for $1/π$ that we want to prove, but avoids the need of solving a system of equations.

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