# 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} }

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 ScienceThe 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 $a=1$

- Mathematics
- 2019

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 ScienceNature
- 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

- Mathematics
- 2014

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

- Mathematics
- 2015

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

- Mathematics
- 2018

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

- Mathematics
- 2017

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

- Mathematics
- 2018

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

- MathematicsElectron. J. Comb.
- 2003

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

- Mathematics
- 1992

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

- Mathematics
- 2010

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

- Mathematics
- 2000

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

- Mathematics
- 2009

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

- Mathematics
- 1985

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

- Mathematics
- 2006

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

- ArtNature
- 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.