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User:Jaume Oliver Lafont/C((d+1)n+d,d)

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\sum_{n=0}^\infty \frac{1}{\binom{3n+2}{2}} = \frac{2\pi}{3\sqrt{3}} = \frac{4}{9}\sin(\frac{\pi}{3})\pi = \frac{\frac{\pi}{3}}{sin(\frac{\pi}{3})}=\frac{1}{sinc(\frac{\pi}{3})} = 1/A086089 [1] Centered nonagonal numbers A060544


\sum_{n=0}^\infty \frac{1}{\binom{4n+3}{3}} = \frac{3}{4}log(4) = \frac{3}{4}(2log(2)) = \frac{3}{4}log(2^2) [2] Odd tetrahedral numbers A015219


\sum_{n=0}^\infty \frac{1}{\binom{5n+4}{4}} = \frac{4}{5}\sqrt{\frac{2}{5}(25-11\sqrt{5})}\pi = \frac{16}{25}(3sin(\frac{2\pi}{5})-4sin(\frac{\pi}{5}))\pi [3] [4] [5] A151989


\sum_{n=0}^\infty \frac{1}{\binom{6n+5}{5}} = \frac{5}{6}(16log(2)-9log(3)) = \frac{5}{6}log(\frac{2^{16}}{3^9}) [6]


\sum_{n=0}^\infty \frac{1}{\binom{7n+6}{6}} = \frac{6}{7}(10\tan(\frac{\pi}{14})-5\tan(\frac{3\pi}{14})+\cot(\frac{\pi}{7}))\pi =\frac{24}{49}(33sin(\frac{\pi}{7})-10sin(\frac{2\pi}{7})-6sin(\frac{3\pi}{7}))\pi =\frac{6}{7}\frac{4}{\sqrt{7}}(3cos(\frac{\pi}{7})+8cos(\frac{2\pi}{7})-2cos(\frac{3\pi}{7})-7)\pi [7] ([8])

\sum_{n=0}^\infty \frac{1}{\binom{8n+7}{7}} = \frac{7}{4}(26log(2)+7\sqrt{2}log(3-2\sqrt{2})) [9]


Contents

Even d

\sum_{n=0}^\infty \frac{1}{\binom{3n+2}{2}} =\frac{2}{3}\cot\left(\frac{\pi}{3}\right)\pi
\sum_{n=0}^\infty \frac{1}{\binom{5n+4}{4}} =\frac{4}{5}\left(\cot\left(\frac{\pi}{5}\right)-3\cot\left(\frac{2\pi}{5}\right)\right)\pi
\sum_{n=0}^\infty \frac{1}{\binom{7n+6}{6}} =\frac{6}{7}\left(\cot\left(\frac{\pi}{7}\right)-5\cot\left(\frac{2\pi}{7}\right)+10\cot\left(\frac{3\pi}{7}\right)\right)\pi

For even values of d, a general expression in terms of alternating sign binomial coefficients and cotangent values follows from the identity


\sum_{n=0}^\infty \left(\frac{1}{Ln+D} - \frac{1}{Ln+(L-D)}\right) = \frac{\pi}{L}\cot(\frac{D\pi}{L})case D=1


(General expression for odd values of d?)

Notes on the digamma function

\Psi(1 - z) - \Psi(z) = \pi\,\!\cot{ \left ( \pi z \right ) }[10],6.3.7. in [11], 6.4.7. in [12]
\Psi(1/2+z) - \Psi(1/2-z) = \pi\,\!\tan { \left (\pi z\right ) }[13] =2\Psi(1+2z)-\Psi(1+z)+\Psi(1-z)-2\Psi(1-2z)\,[14]
\left[\Psi(1 - z) - \Psi(z) \right]\left[ \Psi(1/2+z) - \Psi(1/2-z) \right] = \pi^2 = \left[\Gamma\left(\frac{1}{2}\right)\right]^4

Trigonometric functions defined from Γ(z)

\sin(\pi z) = \frac{\Gamma(\frac{1}{2})^2}{\Gamma(1-z)\Gamma(z)}[15]


\cos(\pi z) = \frac{\Gamma(\frac{1}{2})^2}{\Gamma(\frac{1}{2}+z)\Gamma(\frac{1}{2}-z)}[16][17] =\frac{\Gamma(1-z)\Gamma(z)}{2\Gamma(1-2z)\Gamma(2z)}[18] =\frac{\Gamma(1-z)\Gamma(1+z)}{\Gamma(1-2z)\Gamma(1+2z)} [19]


\tan(\pi z) = \frac{\Gamma(\frac{1}{2}+z)\Gamma(\frac{1}{2}-z)}{\Gamma(1-z)\Gamma(z)} [20]

Sinc function

\frac{\sin(\pi z)}{\pi z} = \frac{1}{\Gamma(1+z)\Gamma(1-z)}[21][22]
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