Catalan's constant

In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link.{{citation

| last = Agol | first = Ian | author-link = Ian Agol

| doi = 10.1090/S0002-9939-10-10364-5

| issue = 10

| journal = Proceedings of the American Mathematical Society

| mr = 2661571

| pages = 3723–3732

| title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds

| volume = 138

| year = 2010| arxiv = 0804.0043| s2cid = 2016662 }}. It is 1/8 of the volume of the complement of the Borromean rings.{{Citation |author=William Thurston|author-link=William Thurston |date=March 2002 |title=The Geometry and Topology of Three-Manifolds |url=http://library.msri.org/books/gt3m/ |chapter=7. Computation of volume |chapter-url=http://library.msri.org/books/gt3m/PDF/7.pdf |archive-url=https://web.archive.org/web/20110125012649/http://library.msri.org/books/gt3m/PDF/7.pdf |archive-date=2011-01-25 |url-status=live |page=165}}

In combinatorics and statistical mechanics, it arises in connection with counting domino tilings,{{citation

| last1 = Temperley | first1 = H. N. V. | author1-link = Harold Neville Vazeille Temperley

| last2 = Fisher | first2 = Michael E. | author2-link = Michael Fisher

| date = August 1961

| doi = 10.1080/14786436108243366

| issue = 68

| journal = Philosophical Magazine

| pages = 1061–1063

| title = Dimer problem in statistical mechanics—an exact result

| volume = 6| bibcode = 1961PMag....6.1061T }} spanning trees,{{citation

| last = Wu | first = F. Y.

| doi = 10.1088/0305-4470/10/6/004

| issue = 6

| journal = Journal of Physics

| mr = 489559

| pages = L113–L115

| title = Number of spanning trees on a lattice

| volume = 10

| year = 1977| bibcode = 1977JPhA...10L.113W

}} and Hamiltonian cycles of grid graphs.{{citation

| last = Kasteleyn | first = P. W. | author-link = Pieter Kasteleyn

| doi = 10.1016/S0031-8914(63)80241-4

| journal = Physica

| mr = 159642

| pages = 1329–1337

| title = A soluble self-avoiding walk problem

| volume = 29

| year = 1963| issue = 12 | bibcode = 1963Phy....29.1329K }}

In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form $n^2+1$ according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of Landau's problems) whether there are even infinitely many primes of this form.{{citation

| last = Shanks | first = Daniel | author-link = Daniel Shanks

| journal = Mathematical Tables and Other Aids to Computation

| mr = 105784

| pages = 78–86

| title = A sieve method for factoring numbers of the form $n^2+1$

| volume = 13

| year = 1959| doi = 10.2307/2001956 | jstor = 2001956 }}

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.{{citation

| last1 = Wyse | first1 = A. B. | author1-link=Arthur Bambridge Wyse

| last2 = Mayall | first2 = N. U.

| bibcode = 1942ApJ....95...24W

| date = January 1942

| doi = 10.1086/144370

| journal = The Astrophysical Journal

| pages = 24–47

| title = Distribution of Mass in the Spiral Nebulae Messier 31 and Messier 33.

| volume = 95| doi-access = free

}}{{citation

| last = van der Kruit | first = P. C.

| bibcode = 1988A&A...192..117V

| date = March 1988

| journal = Astronomy & Astrophysics

| pages = 117–127

| title = The three-dimensional distribution of light and mass in disks of spiral galaxies.

| volume = 192}}

{{unsolved|mathematics|Is Catalan's constant irrational? If so, is it transcendental?}}

It is not known whether {{mvar|G}} is irrational, let alone transcendental.{{citation

| last = Nesterenko | first = Yu. V.

| date = January 2016

| doi = 10.1134/s0081543816010107

| issue = 1

| journal = Proceedings of the Steklov Institute of Mathematics

| pages = 153–170

| title = On Catalan's constant

| volume = 292| s2cid = 124903059

}}. {{mvar|G}} has been called "arguably the most basic constant whose irrationality and transcendence (though strongly

suspected) remain unproven".{{citation

| last1 = Bailey | first1 = David H.

| last2 = Borwein | first2 = Jonathan M.

| last3 = Mattingly | first3 = Andrew

| last4 = Wightwick | first4 = Glenn

| doi = 10.1090/noti1015

| issue = 7

| journal = Notices of the American Mathematical Society

| mr = 3086394

| pages = 844–854

| title = The computation of previously inaccessible digits of $\backslash pi^2$ and Catalan's constant

| volume = 60

| year = 2013| doi-access = free

}}

There exist however partial results. It is known that infinitely many of the numbers β(2n) are irrational, where β(s) is the Dirichlet beta function.{{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://link.springer.com/article/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |hdl=1959.13/803688 |issn=1432-1807}} In particular at least one of β(2), β(4), β(6), β(8), β(10) and β(12) must be irrational, where β(2) is Catalan's constant.{{Cite journal |last=Zudilin |first=Wadim |date=2018-04-26 |title=Arithmetic of Catalan's constant and its relatives |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=89 |pages=45–53 |doi=10.1007/s12188-019-00203-w |arxiv=1804.09922 |language=en}} These results by Wadim Zudilin and Tanguy Rivoal are related to similar ones given for the odd zeta constants ζ(2n+1).

Catalan's constant is known to be an algebraic period, which follows from some of the double integrals given below.

Catalan's constant appears in the evaluation of several rational series including:{{Cite web |last=Weisstein |first=Eric W. |title=Catalan's Constant |url=https://mathworld.wolfram.com/CatalansConstant.html |access-date=2024-10-02 |website=mathworld.wolfram.com |language=en}}$$\backslash frac\{\backslash pi^2\}\{16\}+\backslash frac\; G2\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{(4n+1)^2\}.$$$$\backslash frac\{\backslash pi^2\}\{16\}-\backslash frac\; G2\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{(4n+3)^2\}.$$

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

$$\backslash begin\{align\}$$

G & = 3 \sum_{n=0}^\infty \frac{1}{2^{4n}}

\left(-\frac{1}{2(8n+2)^2}+\frac{1}{2^2(8n+3)^2}-\frac{1}{2^3(8n+5)^2}+\frac{1}{2^3(8n+6)^2}-\frac{1}{2^4(8n+7)^2}+\frac{1}{2(8n+1)^2}\right) \\

& \qquad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}}

\left(\frac{1}{2^4(8n+2)^2}+\frac{1}{2^6(8n+3)^2}-\frac{1}{2^9(8n+5)^2}-\frac{1}{2^{10} (8n+6)^2}-\frac{1}{2^{12} (8n+7)^2}+\frac{1}{2^3(8n+1)^2}\right)

\end{align}

and

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

The theoretical foundations for such series are given by Broadhurst, for the first formula,{{cite arXiv|first1=D. J. |last1=Broadhurst|eprint=math.CA/9803067 |title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {{math|ζ(3)}} and {{math|ζ(5)}}| year=1998}} and Ramanujan, for the second formula.{{cite book|first=B. C.|last=Berndt|title=Ramanujan's Notebook, Part I|publisher=Springer Verlag|date=1985|page=289|isbn=978-1-4612-1088-7}} The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.{{cite journal|first=E. A.| last=Karatsuba| title=Fast evaluation of transcendental functions|journal=Probl. Inf. Transm.|volume=27|issue=4| pages=339–360| date=1991|zbl=0754.65021|mr=1156939}}{{cite book|first=E. A.|last=Karatsuba|contribution=Fast computation of some special integrals of mathematical physics|title=Scientific Computing, Validated Numerics, Interval Methods| url=https://archive.org/details/scientificcomput00wals_919|url-access=limited|editor1-first=W.|editor1-last=Krämer| editor2-first=J. W.|editor2-last=von Gudenberg|pages=[https://archive.org/details/scientificcomput00wals_919/page/n35 29]–41|date=2001|doi=10.1007/978-1-4757-6484-0_3|isbn=978-1-4419-3376-8 }} Using these series, calculating Catalan's constant is now about as fast as calculating Apéry's constant, $\backslash zeta(3)$.

Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:{{cite web|url=http://www.numberworld.org/y-cruncher/internals/formulas.html|title=Formulas and Algorithms|author=Alexander Yee|date=14 May 2019|access-date=5 December 2021}}

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

:$G\; =\; \backslash frac\{1\}\{64\}\backslash sum\_\{k=1\}^\{\backslash infty\; \}\backslash frac\{256^\{k\}(580k^2-184k+15)\}\{k^3(2k-1)\backslash binom\{6k\}\{3k\}\backslash binom\{6k\}\{4k\}\backslash binom\{4k\}\{2k\}\}$

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

All of these series have time complexity $O(n\backslash log(n)^3)$.

As Seán Stewart writes, "There is a rich and seemingly endless source of definite integrals that

can be equated to or expressed in terms of Catalan's constant."{{citation | last = Stewart | first = Seán M. | doi = 10.1017/mag.2020.99 | issue = 561 | journal = The Mathematical Gazette | mr = 4163926 | pages = 449–459 | title = A Catalan constant inspired integral odyssey | volume = 104 | year = 2020| s2cid = 225116026 }} Some of these expressions include:

$$\backslash begin\{align\}$$

G &= -\frac{1}{\pi i}\int_{0}^{\frac{\pi}{2}} \ln\ln \tan x \ln \tan x \,dx \\[3pt]

G &= \iint_{[0,1]^2} \! \frac{1}{1+x^2 y^2} \,dx\, dy \\[3pt]

G &= \int_0^1\int_0^{1-x} \frac{1}{1 -x^2-y^2} \,dy\,dx \\[3pt]

G &= \int_1^\infty \frac{\ln t}{1 + t^2} \,dt \\[3pt]

G &= -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \\[3pt]

G &= \frac{1}{2} \int_0^\frac{\pi}{2} \frac{t}{\sin t} \,dt \\[3pt]

G &= \int_0^\frac{\pi}{4} \ln \cot t \,dt \\[3pt]

G &= \frac{1}{2} \int_0^\frac{\pi}{2} \ln \left( \sec t +\tan t \right) \,dt \\[3pt]

G &= \int_0^1 \frac{\arccos t}{\sqrt{1+t^2}} \,dt \\[3pt]

G &= \int_0^1 \frac{\operatorname{arcsinh} t}{\sqrt{1-t^2}} \,dt \\[3pt]

G &= \frac{1}{2} \int_0^\infty \frac{\operatorname{arctan} t}{t\sqrt{1+t^2}} \,dt \\[3pt]

G &= \frac{1}{2} \int_0^1 \frac{\operatorname{arctanh} t}{\sqrt{1-t^2}} \,dt \\[3pt]

G &= \int_0^\infty \arccot e^{t} \,dt \\[3pt]

G &= \frac{1}{4} \int_0^{{\pi^2}/{4}} \csc \sqrt{t} \,dt \\[3pt]

G &= \frac{1}{16} \left(\pi^2 + 4\int_1^\infty \arccsc^2 t \,dt\right) \\[3pt]

G &= \frac{1}{2} \int_0^\infty \frac{t}{\cosh t} \,dt \\[3pt]

G &= \frac{\pi}{2} \int_1^\infty \frac{\left(t^4-6t^2+1\right)\ln\ln t}{\left(1+t^2\right)^3} \,dt \\[3pt]

G &= \frac{1}{2} \int_0^\infty \frac{\arcsin \left(\sin t\right)}{t} \,dt \\[3pt]

G &= 1 + \lim_{\alpha\to{1^-}}\!\left\{\int_0^{\alpha}\!\frac{\left(1+6t^2+t^4\right)\arctan{t}}{t\left(1-t^2\right)^2}\, dt

+ 2\operatorname{artanh}{\alpha} - \frac{\pi\alpha}{1-\alpha^2} \right\} \\[3pt]

G &= 1 - \frac18 \iint_{\R^2}\!\!\frac{x\sin\left(2xy/\pi\right)}{\,\left(x^2+\pi^2\right)\cosh x\sinh y\,} \,dx\,dy \\[3pt]

G &= \int_{0}^{\infty}\int_{0}^{\infty}\frac{\sqrt[4]{x} \left(\sqrt{x} \sqrt{y}-1\right)}{(x+1)^2 \sqrt[4]{y} (y+1)^2 \log (x y)}dxdy

\end{align}

where the last three formulas are related to Malmsten's integrals.{{Cite journal| first1=Iaroslav| last1=Blagouchine| title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results| year=2014| doi=10.1007/s11139-013-9528-5| url=https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf| volume=35| journal=The Ramanujan Journal| pages=21–110| s2cid=120943474| access-date=2018-10-01| archive-url=https://web.archive.org/web/20181002020243/https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf| archive-date=2018-10-02| url-status=dead }}

If {{math|K(k)}} is the complete elliptic integral of the first kind, as a function of the elliptic modulus {{math|k}}, then

$$G\; =\; \backslash tfrac\{1\}\{2\}\; \backslash int\_0^1\; \backslash mathrm\{K\}(k)\backslash ,dk$$

If {{math|E(k)}} is the complete elliptic integral of the second kind, as a function of the elliptic modulus {{math|k}}, then

$$G\; =\; -\backslash tfrac\{1\}\{2\}+\backslash int\_0^1\; \backslash mathrm\{E\}(k)\backslash ,dk$$

With the gamma function {{math|1=Γ(x + 1) = x!}}

$$\backslash begin\{align\}$$

G &= \frac{\pi}{4} \int_0^1 \Gamma\left(1+\frac{x}{2}\right)\Gamma\left(1-\frac{x}{2}\right)\,dx \\

&= \frac{\pi}{2} \int_0^\frac12\Gamma(1+y)\Gamma(1-y)\,dy

\end{align}

The integral

$$G\; =\; \backslash operatorname\{Ti\}\_2(1)=\backslash int\_0^1\; \backslash frac\{\backslash arctan\; t\}\{t\}\backslash ,dt$$

is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

{{mvar|G}} appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:

$$\backslash begin\{align\}$$

\psi_1 \left(\tfrac14\right) &= \pi^2 + 8G \\

\psi_1 \left(\tfrac34\right) &= \pi^2 - 8G.

\end{align}

Simon Plouffe gives an infinite collection of identities between the trigamma function, {{pi}}² and Catalan's constant; these are expressible as paths on a graph.

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes G-function, as well as integrals and series summable in terms of the aforementioned functions.

As a particular example, by first expressing the inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes {{mvar|G}}-function, the following expression is obtained (see Clausen function for more):

$$G=4\backslash pi\; \backslash log\backslash left(\; \backslash frac\{\; G\backslash left(\backslash frac\{3\}\{8\}\backslash right)\; G\backslash left(\backslash frac\{7\}\{8\}\backslash right)\; \}\{\; G\backslash left(\backslash frac\{1\}\{8\}\backslash right)\; G\backslash left(\backslash frac\{5\}\{8\}\backslash right)\; \}\; \backslash right)\; +4\; \backslash pi\; \backslash log\; \backslash left(\; \backslash frac\{\; \backslash Gamma\backslash left(\backslash frac\{3\}\{8\}\backslash right)\; \}\{\; \backslash Gamma\backslash left(\backslash frac\{1\}\{8\}\backslash right)\; \}\; \backslash right)\; +\backslash frac\{\backslash pi\}\{2\}\; \backslash log\; \backslash left(\; \backslash frac\{1+\backslash sqrt\{2\}\; \}\{2\; \backslash left(2-\backslash sqrt\{2\}\backslash right)\}\; \backslash right).$$

If one defines the Lerch transcendent {{math|Φ(z,s,α)}} by

$$\backslash Phi(z,\; s,\; \backslash alpha)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\; \{\; z^n\}\; \{(n+\backslash alpha)^s\},$$

then

$$G\; =\; \backslash tfrac\{1\}\{4\}\backslash Phi\backslash left(-1,\; 2,\; \backslash tfrac\{1\}\{2\}\backslash right).$$

{{mvar|G}} can be expressed in the following form:{{cite journal | journal=Acta Arithmetica |volume=103 |issue=4 |pages=329–342 | author=Bowman, D. | author2=Mc Laughlin, J.| name-list-style=amp | title=Polynomial continued fractions | language=English | year=2002 |doi=10.4064/aa103-4-3 |arxiv=1812.08251 |bibcode=2002AcAri.103..329B |s2cid=119137246 | url=https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/4paper1.pdf |archive-url=https://web.archive.org/web/20200413012537/https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/4paper1.pdf |archive-date=2020-04-13 |url-status=live}}

:$G=\backslash cfrac\{1\}\{1+\backslash cfrac\{1^4\}\{8+\backslash cfrac\{3^4\}\{16+\backslash cfrac\{5^4\}\{24+\backslash cfrac\{7^4\}\{32+\backslash cfrac\{9^4\}\{40+\backslash ddots\}\}\}\}\}\}$

The simple continued fraction is given by:{{Cite web |title=A014538 - OEIS |url=http://oeis.org/A014538 |access-date=2022-10-27 |website=oeis.org}}

:$G=\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{10+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{8+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{88+\backslash ddots\}\}\}\}\}\}$

This continued fraction would have infinite terms if and only if $G$ is irrational, which is still unresolved.

The number of known digits of Catalan's constant {{mvar|G}} has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.{{cite web|last1=Gourdon|first1=X.|last2=Sebah|first2=P.|url=http://numbers.computation.free.fr/Constants/constants.html|title=Constants and Records of Computation|access-date=11 September 2007}}

• Gieseking manifold
• List of mathematical constants
• Mathematical constant
• Particular values of Riemann zeta function

{{reflist}}

• {{cite journal

|first = Victor

|last = Adamchik

|year = 2002

|journal = Zeitschrift für Analysis und ihre Anwendungen

|volume = 21

|issue = 3

|pages = 1–10

|title = A certain series associated with Catalan's constant

|mr = 1929434

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|doi-access = free

}}

• {{cite conference

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| editor1-last = Watanabe | editor1-first = Shunro

| editor2-last = Nagata | editor2-first = Morio

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| doi = 10.1145/96877.96917

| isbn = 0201548925

| s2cid = 1949187

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| year = 1990| doi-access = free

}}

• {{Cite journal

| first = David M.

| last = Bradley

| title = A class of series acceleration formulae for Catalan's constant

| doi = 10.1023/A:1006945407723

| year = 1999

| journal = The Ramanujan Journal

| volume = 3

| issue = 2

| pages = 159–173

| mr = 1703281

| arxiv = 0706.0356

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| s2cid = 5111792

}}

• {{cite web

| first = Victor

| last = Adamchik

| url = http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm

| title = 33 representations for Catalan's constant

| archiveurl = https://web.archive.org/web/20160807111945/https://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm

| archivedate = 2016-08-07

| access-date = 14 July 2005

}}

• {{cite web

| first = Simon

| last = Plouffe

| url = http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html

| title = A few identities (III) with Catalan

| year = 1993

| archiveurl = https://web.archive.org/web/20190626124124/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html

| archivedate = 2019-06-26

| access-date = 29 July 2005

}} (Provides over one hundred different identities).

• {{cite web

| first = Simon

| last = Plouffe

| url = http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html

| title = A few identities with Catalan constant and Pi^2

| year = 1999

| archiveurl = https://web.archive.org/web/20190626124128/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html

| archivedate = 2019-06-26

| access-date = 29 July 2005

}} (Provides a graphical interpretation of the relations)

• {{cite book

| first = Greg

| last = Fee

| url = https://www.gutenberg.org/ebooks/682

| title = Catalan's Constant (Ramanujan's Formula)

| year = 1996

}} (Provides the first 300,000 digits of Catalan's constant)

• {{citation

| first = David M.

| last = Bradley

| title = Representations of Catalan's constant

| citeseerx = 10.1.1.26.1879

| year = 2001

| url = https://www.researchgate.net/publication/2325473

| mode = cs1

}}

• {{cite web | first = Fredrik | last = Johansson |url = https://fungrim.org/ordner/0.915965594177219015054603514932/

| title = 0.915965594177219015054603514932 | work = Ordner, a catalog of real numbers in Fungrim | accessdate= 21 April 2021 }}

• {{cite web

| title = Catalan's Constant

| url = https://www.youtube.com/watch?v=e5wqw2_EkxQ&list=PLW1_9UnhaSkGqlwbQphLMGCx2JvaGu1HB&index=72

| publisher = Let's Learn, Nemo!

| date = 10 August 2020

| website = YouTube

| access-date= 6 April 2021

}}

• {{MathWorld |title = Catalan's Constant |urlname = CatalansConstant}}
• {{WolframFunctionsSite |title = Catalan constant: Series representations |urlname = Constants/Catalan/06/01/}}
• {{springer |title = Catalan constant |id = p/c130040 | mode=cs1}}

Category:Combinatorics

Category:Mathematical constants