$\newcommand{\mt}[4]{\begin{bmatrix}#1\\#3\end{bmatrix}}$ Via jupytext this file can be shown as a jupyter notebook.
Gosper family of $\frac{1}{\pi}$ formulas¶
This file computes some Ramanujan-Sato series for $\frac{1}{\pi}$ of the Gosper family, i.e. for Sato triples of the form \begin{align} (N, \gamma_N, \tau_N) &= \left( N, \mt{2N}{-2}{2N+1}{-2}, \frac{2}{2N+1} \frac{2N+i\sqrt{2N}}{2N} \right). \end{align}
In [1]:
)cd ..
)read input/jfricas-test-support.input )quiet
The current FriCAS default directory is /home/hemmecke/backup/git/qeta All user variables and function definitions have been cleared. All )browse facility databases have been cleared. Internally cached functions and constructors have been cleared. )clear completely is finished. The current FriCAS default directory is /home/hemmecke/backup/git/qeta/tmp
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)set output formatted off
)set output algebra on
Setup¶
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-------------------------------------------------------------------
--setup
-------------------------------------------------------------------
In [3]:
)set stream calculate 2
C ==> QQ
MZZ ==> Matrix ZZ
QCMODPOL ==> QEtaClassicalModularPolynomial
AN ==> AlgebraicNumber
SATOTRI ==> Record(fnn: PP, fgamma: MZZ, ftau: AN, ftaufd: AN)
QETAAUX ==> QEtaAuxiliaryPackage
Px ==> UP('x, ZZ)
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digits 100;
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)read projectdir.input )quiet
In [6]:
gammaFD tau ==> toFundamentalDomain(tau, 2)
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oopiG(nn) ==> (_
gammaN := matrix [[2*nn,-2],[2*nn+1,-2]];_
tauN := (2*nn+sqrt(-2*nn))/(nn*(2*nn+1));_
gammafd := gammaFD tauN;_
taufd := moebiusTransform(gammafd, tauN);_
basedir := PROJECTDIR "/data/oneoverpi/gosper";_
modpoldir: String := basedir "/" string(nn);_
iotaAtTauN := gosperIotaAt(nn,1,taufd,basedir);_
zN := simplifyRadicals(1/iotaAtTauN);_
gosperOneOverPiFormula(nn,gammaN,tauN,zN,basedir))
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-------------------------------------------------------------------
--endsetup
-------------------------------------------------------------------
Gosper family¶
In [9]:
oopiG 3
Out[9]:
\[
\frac{1}{\pi }=\frac{\sqrt{3}}{6}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{2304}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(8\, n+1\right)}\right)
\]
In [10]:
oopiG 5
Out[10]:
\[
\frac{1}{\pi }=\frac{2\, \sqrt{2}}{9}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{20736}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(10\, n+1\right)}\right)
\]
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oopiG 7
Out[11]:
\[
\frac{1}{\pi }=\frac{\sqrt{7}\, \sqrt{22\, \sqrt{2}-25}}{98}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{-176\, \sqrt{2}+249}{12544}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(56\, n-3\, \sqrt{2}+9\right)}\right)
\]
In [12]:
oopiG 11
Out[12]:
\[
\frac{1}{\pi }=\frac{\sqrt{11}}{198}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{2509056}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(280\, n+19\right)}\right)
\]
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oopiG 13
Out[13]:
\[
\frac{1}{\pi }=\frac{\left(-11001958\, {\sqrt[3]{3}}^{2}\, \sqrt{39}-10142600\, \sqrt{2}\, {\sqrt[3]{3}}^{2}\right)\, {\sqrt[3]{4807\, \sqrt{78}+450474813}}^{2}+\left(12163982446\, \sqrt[3]{3}\, \sqrt{39}-11218700766\, \sqrt{2}\, \sqrt[3]{3}\right)\, \sqrt[3]{4807\, \sqrt{78}+450474813}+187762611676731\, \sqrt{2}}{5069511696287989410312586958712048675114}\, \left(\sum_{n=0}^{\infty }{\frac{\left(4\, n\right)!}{{n!}^{4}}\, {\left(\frac{\left(1814046432\, {\sqrt[3]{3}}^{2}\, \sqrt{78}+657072736\, {\sqrt[3]{3}}^{2}\right)\, {\sqrt[3]{4807\, \sqrt{78}+450474813}}^{2}+\left(-2005613919232\, \sqrt[3]{3}\, \sqrt{78}+728127660960\, \sqrt[3]{3}\right)\, \sqrt[3]{4807\, \sqrt{78}+450474813}-1608184278790887}{5600150878847019264}\right)}^{n}\, \left(\left(\left(-4132438706990\, {\sqrt[3]{3}}^{2}\, \sqrt{39}+14861487699362\, \sqrt{2}\, {\sqrt[3]{3}}^{2}\right)\, {\sqrt[3]{4807\, \sqrt{78}+450474813}}^{2}+\left(4568465440883498\, \sqrt[3]{3}\, \sqrt{39}+16428930176758764\, \sqrt{2}\, \sqrt[3]{3}\right)\, \sqrt[3]{4807\, \sqrt{78}+450474813}-36328986949095263157\, \sqrt{2}\right)\, \sqrt{\left(-51120003453348698\, {\sqrt[3]{3}}^{2}\, \sqrt{78}+7185856238296904823425\, {\sqrt[3]{3}}^{2}\right)\, {\sqrt[3]{4807\, \sqrt{78}+450474813}}^{2}+\left(-28259087496316379549\, \sqrt[3]{3}\, \sqrt{78}+7944668476360086351891633\, \sqrt[3]{3}\right)\, \sqrt[3]{4807\, \sqrt{78}+450474813}+8783609797976970042266458026}+112163545376868390195294504\, n\right)}\right)
\]
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oopiG 17
Out[14]:
\[
\frac{1}{\pi }=\frac{\sqrt{34}\, \sqrt{33\, \sqrt{17}-131}}{7956}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{-528\, \sqrt{17}+2177}{20736}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(3536\, n-81\, \sqrt{17}+527\right)}\right)
\]
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oopiG 19
Out[15]:
\[
\frac{1}{\pi }=\frac{\sqrt{399}\, \sqrt{\left(-97954978\, {\sqrt[3]{2}}^{2}\, \sqrt{114}-241027648\, {\sqrt[3]{2}}^{2}\right)\, {\sqrt[3]{1023\, \sqrt{114}+992484832}}^{2}+\left(123105910400\, \sqrt[3]{2}\, \sqrt{114}-302927634308\, \sqrt[3]{2}\right)\, \sqrt[3]{1023\, \sqrt{114}+992484832}+7824834886206821}}{234354416037829557247170635312994}\, \left(\sum_{n=0}^{\infty }{\frac{\left(4\, n\right)!}{{n!}^{4}}\, {\left(\frac{\left(19590995600\, {\sqrt[3]{2}}^{2}\, \sqrt{114}+48205529600\, {\sqrt[3]{2}}^{2}\right)\, {\sqrt[3]{1023\, \sqrt{114}+992484832}}^{2}+\left(-24621182080000\, \sqrt[3]{2}\, \sqrt{114}+60585526861600\, \sqrt[3]{2}\right)\, \sqrt[3]{1023\, \sqrt{114}+992484832}-152278733339406331}{361648190438901214464}\right)}^{n}\, \left(\left(\left(-3658078\, {\sqrt[3]{2}}^{2}\, \sqrt{266}+92791535\, {\sqrt[3]{2}}^{2}\, \sqrt{21}\right)\, {\sqrt[3]{1023\, \sqrt{114}+992484832}}^{2}+\left(4596954299\, \sqrt[3]{2}\, \sqrt{266}+116616225292\, \sqrt[3]{2}\, \sqrt{21}\right)\, \sqrt[3]{1023\, \sqrt{114}+992484832}-177741988203885\, \sqrt{21}\right)\, \sqrt{\left(-97954978\, {\sqrt[3]{2}}^{2}\, \sqrt{114}-241027648\, {\sqrt[3]{2}}^{2}\right)\, {\sqrt[3]{1023\, \sqrt{114}+992484832}}^{2}+\left(123105910400\, \sqrt[3]{2}\, \sqrt{114}-302927634308\, \sqrt[3]{2}\right)\, \sqrt[3]{1023\, \sqrt{114}+992484832}+7824834886206821}+860538537851467974043160\, n\right)}\right)
\]
In [16]:
oopiG 23
Out[16]:
\[
\frac{1}{\pi }=\frac{\sqrt{23}\, \sqrt{78\, \sqrt{2}-1}}{9522}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{-30576\, \sqrt{2}+43241}{10969344}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(1288\, n-81\, \sqrt{2}+175\right)}\right)
\]
In [17]:
oopiG 29
Out[17]:
\[
\frac{1}{\pi }=\frac{2\, \sqrt{2}}{9801}\, \left(\sum_{n=0}^{\infty }{{\left(\frac{1}{24591257856}\right)}^{n}\, \frac{\left(4\, n\right)!}{{n!}^{4}}\, \left(26390\, n+1103\right)}\right)
\]
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