# On Catalan’S constant

@article{Nesterenko2016OnCC,
title={On Catalan’S constant},
author={Yu. V. Nesterenko},
journal={Proceedings of the Steklov Institute of Mathematics},
year={2016},
volume={292},
pages={153-170}
}
• Y. Nesterenko
• Published 14 May 2016
• Mathematics
• Proceedings of the Steklov Institute of Mathematics
A new efficient construction of Diophantine approximations to Catalan’s constant is presented that is based on the direct analysis of the representation of a hypergeometric function with specially chosen half-integer parameters as a series and as a double Euler integral over the unit cube. This allows one to significantly simplify the proofs of Diophantine results available in this domain and substantially extend the capabilities of the method. The sequences of constructed rational…
9 Citations
On the Connection Between Irrationality Measures and Polynomial Continued Fractions
• Mathematics
• 2021
Linear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence Fn = Fn−1 + Fn−2, have been intensely studied over millennia and yet still hide interesting
Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry
• Philosophy, Computer Science
The Ramanujan Journal
• 2022
As we all know, he was proven wrong by Gödel and Turing in general, but even for such concrete problems, like the irrationality of a specific, natural, constant, like the Euler-Mascheroni constant
Normality Analysis of Current World Record Computations for Catalan's Constant and Arc Length of a Lemniscate with $a=1$
Catalan's constant and the lemniscate constants have been important mathematical constants of interest to the mathematical society, yet various properties are unknown. An important property of
Generating conjectures on fundamental constants with the Ramanujan Machine.
• Mathematics, Computer Science
Nature
• 2021
This work supports a different conceptual framework for research: computer algorithms use numerical data to unveil mathematical structures, thus trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research.
Hankel and Toeplitz Determinants
det(Hn) = Qn−1 k=1(k!) 4 Q2n−1 =1 ! = G(n+ 1) G(2n+ 1) → 0 as n→∞. More precisely [1], det(Hn) 4−n(2π)nn−1/4 → 21/12e1/4A−3 = 0.6450024485... where A denotes the Glaisher-Kinkelin constant [2]. Such
A Determinantal Approach to Irrationality
It is a classical fact that the irrationality of a number $$\xi \in \mathbb R$$ξ∈R follows from the existence of a sequence $$p_n/q_n$$pn/qn with integral $$p_n$$pn and $$q_n$$qn such that q_n\xi
Maximum Queue Length for Traffic Light with Bernoulli Arrivals
Cars arrive at an intersection with a stoplight, which is either red or green. The cars all travel in the same direction, that is, we ignore cross-traffic & oncoming traffic. Assume that the
Triangles Formed via Poisson Nearest Neighbors
We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.
How Far Might We Walk at Random
This elementary treatment first summarizes extreme values of a Bernoulli random walk on the one-dimensional integer lattice over a finite discrete time interval. Both the symmetric (unbiased) and

## References

SHOWING 1-10 OF 21 REFERENCES
An Apery-like Difference Equation for Catalan's Constant
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 coecients, a second-order dierence equation is obtained and a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it is derived.
Diophantine properties of numbers related to Catalan's constant
• Mathematics
• 2003
1) as a linear form in 1 and even beta valuesonly. The following Lemma 2 is almost the same as the correspondingone in [BR] and we only include a sketch of the proof. We would like toemphasise the
ON ARITHMETIC PROPERTIES OF THE VALUES OF HYPERGEOMETRIC FUNCTIONS
The author proposes an effective method of constructing a linear approximating form for a hypergeometric function of general type and its derivatives, which has a zero of the maximal possible order
On the irrationality exponent of the number ln 2
We propose another method of deriving the Marcovecchio estimate for the irrationality measure of the number ln 2 following, for the most part, the method of proof of the irrationality of the number
Lower bounds for linear forms in values of certain hypergeometric functions
By using Pade approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are
Rational approximations for values of derivatives of the Gamma function
The arithmetic nature of Euler's constant γ is still unknown and even getting good rational approximations to it is difficult. Recently, Aptekarev managed to find a third order linear recurrence with
Ramanujan's Notebooks
During the time period between 1903 and 1914, Ramanujan worked in almost complete isolation in India. Throughout these years, he recorded his mathematical results without proofs in notebooks. Upon
Nombres D’Euler, approximants de Padé et constante de Catalan
RésuméAu moyen d’une méthode d’approximation de Padé introduite par Prévost dans [13], nous construisons des familles d’approximations rationnelles rapidement convergentes vers la constante de
A Course of Modern Analysis
• Art
Nature
• 1916
The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.