arXiv subjects
Publications and source records attributed to Jesús Guillera.
Using a self-replicating method, we generalize with a free parameter some Borwein algorithms for the number $π$. This generalization includes values of the Gamma function like $Γ(1/3)$, $Γ(1/4)$ and of course $Γ(1/2)=\sqrtπ$. In addition, we give new rapid algorithms for the perimeter of an ellipse.
Special arithmetic series $f(x)=\sum_{n=0}^{\infty}c_nx^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy "self-replicating" functional identities. For example, the equation $$\frac1{(1+4z)^2}f\biggl(\frac{z}{(1+4z)^3}\biggr)=\frac1{(1+2z)^2}f\biggl(\frac{z^2}{(1+2z)^3}\biggr)$$ generates a modular form $f(x)$ of weight 2 and level 7, when a related modular parametrization $x=x(τ)$ is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing $π$ and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.
We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/π$. We will not use the modular theory nor either the Gauss' formula that we used in another paper. Our proofs are short and self-contained.
We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cramér, and also new representations for some arithmetical functions in terms of the non-trivial zeros.
We explain an experimental method to find CY-type differential equations of order $3$ related to modular functions of genus zero. We introduce a similar class of Calabi-Yau differential equations of order $5$, show several examples and make a conjecture related to some geometric invariants. We finish the paper with a few examples of seven order.
We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lalín, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper "Periods".
We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/π$. The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate non-singular points into singular ones, where the required constants can be computed using asymptotic analysis.
Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/π^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new examples of series of non-hypergeometric type. To motivate our theory we begin with the simpler case of Ramanujan-Sato series for $1/π$.
We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/π^2.
We prove that there is a correspondence between Ramanujan-type formulas for 1/π, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.
Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for $1/π^2$ of hypergeometric and non-hypergeometric types.
"Divergent" Ramanujan-type series for $1/π$ and $1/π^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique.
We generalize the patterns of supercongruences of Ramanujan-type observed by L. Van Hamme and W. Zudilin to series involving simple square roots anywhere and not only in the result of the sum. To support our observations we give some examples.
Our main results are a WZ-proof of a new Ramanujan-like series for $1/π^2$ and a hypergeometric identity involving three series.