Catalan's constant

The Catalan constant , commonly referred to as, is a mathematical constant . She is the value of the series ${\displaystyle G}$

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}=1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+\cdots ,}$

thus the value of the Dirichlet beta function at position 2. The constant is named after Eugène Catalan . Their irrationality is presumed, but is still unproven today. It is known that an infinite number of the numbers must be irrational, with at least one of and . ${\displaystyle \beta (2)}$ ${\displaystyle \beta (2k),k=1,2,3,\ldots }$ ${\displaystyle \beta (2),\beta (4),\beta (6),\beta (8),\beta (10),\beta (12)}$ ${\displaystyle \beta (14)}$

History and title

Catalan referred to this constant in a work from 1867 and gave numerous integral and series representations for it. ${\displaystyle G}$

value

Is an approximation

${\displaystyle G=0,91596\ 55941\ 77219\ 01505\ 46035\ 14932\ 38411\ 07741\ 49374\ 28167\ \dots }$(Follow A006752 in OEIS )

Currently (February 28, 2020), according to a calculation by Seungmin Kim from July 16, 2019, 600,000,000,000 decimal places are known.

Further representations

There is an abundance of other representations, a fraction of which are given below:

Integral representations

${\displaystyle G=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,{\rm {d}}t}$

${\displaystyle G=\int _{0}^{1}{\frac {\arctan t}{t}}\,{\rm {d}}t}$

${\displaystyle G=\int _{0}^{1}\!\!\int _{0}^{1}{\frac {1}{1+x^{2}y^{2}}}\,{\rm {d}}x\,{\rm {d}}y}$

Series representations

According to S. Ramanujan :

${\displaystyle G={\frac {\pi }{8}}\ln \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}\ .}$

Another series contains the Riemann zeta function :

${\displaystyle G={\frac {1}{16}}\sum _{n=1}^{\infty }(n+1){\frac {3^{n}-1}{4^{n}}}\zeta (n+2)\ .}$

The following sum converges very quickly ( Alexandru Lupaş 2000):

${\displaystyle G={\frac {1}{64}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}\cdot 2^{8n}\cdot (40n^{2}-24n+3)\cdot (2n)!^{3}\cdot n!^{2}}{n^{3}\cdot (2n-1)\cdot (4n)!^{2}}}\ .}$

According to Jesus Guillera , the following series are valid, which converge faster than the series by Lupaş :

${\displaystyle G={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom {2k}{k}}^{3}}}}$,

${\displaystyle G=-{\frac {1}{1024}}\sum _{k=1}^{\infty }{\frac {(-4096)^{k}\left(45136k^{4}-57184k^{3}+21240k^{2}-3160k+165\right)}{k^{3}(2k-1)^{3}}}\left({\frac {(2k)!^{6}(3k)!^{3}}{k!^{3}(6k)!^{3}}}\right)}$.

According to Pilehrood , the following series apply, which also converge faster than the series from Lupaş :

${\displaystyle G={\frac {1}{64}}\sum _{k=1}^{\infty }{\frac {256^{k}\left(580k^{2}-184k+15\right)}{k^{3}(2k-1){\binom {6k}{3k}}{\binom {6k}{4k}}{\binom {4k}{2k}}}}}$,

${\displaystyle G=-{\frac {1}{64}}\sum _{k=1}^{\infty }{\frac {(-256)^{k}\left(419840k^{6}-915456k^{5}+782848k^{4}-332800k^{3}+73256k^{2}-7800k+315\right)}{k^{3}(2k-1)(4k-1)^{2}(4k-3)^{2}{\binom {8k}{4k}}^{2}{\binom {2k}{k}}}}}$.

BBP-like series

A BBP series has been looking for a long time . At first only very long specimens were found. The 9-part by Victor Adamchik (2007) is relatively short :

${\displaystyle {\textstyle G={\frac {3}{64}}\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}}{64^{n}}}\left({\frac {32}{(12n+1)^{2}}}-{\frac {32}{(12n+2)^{2}}}-{\frac {32}{(12n+3)^{2}}}-{\frac {8}{(12n+5)^{2}}}-{\frac {16}{(12n+6)^{2}}}-{\frac {4}{(12n+7)^{2}}}-{\frac {4}{(12n+9)^{2}}}-{\frac {2}{(12n+10)^{2}}}+{\frac {1}{(12n+11)^{2}}}\right)}}$

literature

• E. Catalan : Mémoire sur la transformation des séries et sur quelques intégrales définies (April 1, 1865), Mémoires couronnés et mémoires des savants étrangers 33, 1867, pp. 1-50 (French; "G = 0.915 965 594 177 21" on p. 30; in the Internet archive: [1] )
• LA Ljusternik : Mathematical Analysis. Functions, Limits, Series, Continued Fractions , 1965, pp. 313-314 (English)

Individual evidence

1. ↑ Tanguy Rivoal, Wadim Zudilin: Diophantine properties of numbers related to Catalan's constant ( PDF file, 207 kB), Mathematische Annalen 326, August 2003, pp. 705–721 (English).
2. ↑ Alexander Yee: Records set by y-cruncher. August 24, 2017, accessed on February 28, 2020 . 
3. ↑ Alexandru Lupaş : Formulas for some classical constants ( PDF file, 169 kB), Preprint, 2000; in Heiner Gonska et al. (Ed.): Proceedings of the 4th Romanian-German seminar on approximation theory and its applications, Braşov, Romania, July 3-5, 2000 , series of publications from the Mathematics Department of the Gerhard Mercator University Duisburg SM-DU-485, 2000, Pp. 70-76.
4. ↑ ^{a b c} Alexander J. Yee: Formulas and Algorithms. Retrieved March 15, 2020 . 
5. ↑ Jesus Guillera: a new formula for computing the Catalan constant. Retrieved March 15, 2020 . 
6. ↑ Jesus Guillera: Hypergeometric Identities for 10 extended Ramanujan-type series . arxiv : 1104.0396v1 .
7. ↑ Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood: Series acceleration formulas for beta values . In: Discrete Mathematics and Theoretical Computer Science . tape 12 , no. 2 , 2010, p. 223-236 ( inria.fr ).

Web links

• Eric W. Weisstein : Catalan's Constant . In: MathWorld (English).
• Catalan constant at The Wolfram Functions Site (English; with calculation option)
• Sequence A014538 in OEIS ( continued fraction expansion of G )
• Episode A054543 in OEIS (Engel development of G )
• Episode A132201 in OEIS (Pierce development of G )