Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we obtain a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it. Similar arguments can be put forward to indicate a second-order difference equation and a new continued fraction for $ζ(4)=π^4/90$, and we announce corresponding results at the end of this paper.
In this paper, Riemann's Zeta function with odd positive integer argument is represented as an infinite summation of integer powers of $π$ with rational coefficients. Specific values for Apery's Constant and Catalan's Constant are then derived.
We prove that at least one of the six numbers $β(2i)$ for $i=1,\dots,6$ is irrational. Here $β(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes Dirichlet's beta function, so that $β(2)$ is Catalan's constant.
In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalan's constant and pi times the logarithm of an algebraic unit.
It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8ϕ_7$-series and a balanced $_4ϕ_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan's constant. This proof is considerably simpler and more stream-lined than the first proof, due to the second author.
In the joint work of T.Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function $β(s)$ at even positive integers. The construction gives some bonuses for Catalan's constant $G=β(2)$, such as a second-order Apery-like recursion and a permutation group in the sense of G.Rhin and C.Viola. Here we prove expected integrality properties of solutions to the above recursion as well as suggest a simpler (also second-order and Apery-like) one for $G$. We "enlarge" the permutation group of our previous (joint with T.Riv…
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.
The Ramanujan Machine project detects new expressions related to constants of interest, such as $ζ$ function values, $γ$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the Catalan constant $G= 0.91596559\ldots$ We show how to generate infinitely many. We used an ad hoc software toolchain and rather tedious mathematical developments. Because we do not provide a proper peer-reviewed proof of the relations given here we do not claim them to be theorems.
In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving zeta(3), pi, and Catalan's constant. They indicate that they were not able to deduce these products directly from the values of a function introduced in 1993 by Borwein and Dykshoorn. We provide here such a proof for two of these formulas. We also give a direct proof for the other two formulas, by using a generalization of the Borwein-Dykshoorn function due to Adamchik. Finally we give an expression of the Borwein-Dykshoorn function in terms of the "parameterized-Euler-constant function" introduced b…
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $π^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.
A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional integral and several special cases are deduced. On the way, we also highlight a relationship with some multivariate generalizations of the Riemann zeta function.
We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, …
In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $β{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $ζ(3)$, and the Catalan constant, $G = β(2)$. Interestingly, our series for $\,ζ(3)\,$ converges faster than that used…
