美しすぎる二項係数付き級数一覧
Notation
\begin{align} \beta_x\coloneqq\frac{\Gamma(x+\frac{1}{2})}{\Gamma(\frac{1}{2})\Gamma(x+1)}, \qquad \beta_n=\frac{\binom{2n}{n}}{2^{2n}},\qquad \beta_{n+\frac12}=\frac{2^{2n}}{\pi(n+\frac12)\binom{2n}{n}} \end{align}
\begin{align}
\beta_x^{r,k}\coloneqq\gamma_{r,k}(x)\beta_{x}^r
\end{align}
\begin{align} &\gamma_{2r,1}(x)=\pi(2x+\frac12),&\gamma_{2r+1,-1}(x)=(-1)^{\lfloor x\rfloor}\pi(2x+\frac12)\\ &\gamma_{5,-3}(x)=(-1)^{\lfloor x\rfloor}\pi^3\frac{(4x+1)^3+(4x+1)}{16}\\ & \gamma_{3,1}(x)=\pi^2(x+\frac14) \end{align}
\begin{align} G=\sum_{0\leq n}\frac{(-1)^n}{(2n+1)^2},\qquad A=\sum_{0\leq n}\beta_n^3=\frac{\pi}{\Gamma\left(\frac34\right)^4} \end{align}
\begin{align} \sum_{n\leq k}\beta_{k+\frac12}^{5,-3}\coloneqq\sum_{0\leq k}\left(\beta_{k+\frac12}^{5,-3}-2\beta_k^{1,-1}\right)-\sum_{k=0}^{n-1}\beta_{k+\frac12}^{5,-3} \end{align}
Series
\begin{align}
\ \sum_{0\leq n}\beta_n^3\left(\frac{8G}{\pi^2}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^2\right)&=\sum_{0\leq n}\beta_n^3 \\
\sum_{0\leq n}(-1)^n\beta_n^3\left(\frac{8G}{\pi^2}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^2\right)&=\frac{1}{\sqrt{2}}\sum_{0\leq n}(-1)^n\beta_n^3
\end{align}
\begin{align}
\sum_{0\leq n}\beta_{n+\frac{1}{2}}^3\sum_{k=0}^n\beta_k^{4,1}&=\sum_{0\leq n}\beta_n^3\\
\sum_{0\leq n}(-1)^n\beta_{n+\frac{1}{2}}^3\sum_{k=0}^n\beta_k^{4,1}&=\frac{1}{3}\sum_{0\leq n}(-1)^n\beta_n^3
\end{align}
\begin{align} \sum_{0\leq n }\beta_{n+\frac{1}{2}}^{3,-1}=\frac{8G}{\pi^2} \end{align}
\begin{align}
\sum_{0\leq n}\left(\beta_{n+\frac{1}{2}}^{4,1}-\frac{2}{\pi}\frac{1}{n+1}\right)=\frac{6\ln2}{\pi}-\frac{28\zeta(3)}{\pi^3}
\end{align}
\begin{align}
\sum_{0\leq n}\left(\beta_{n+\frac{1}{2}}^{4,1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^2-\frac{2}{\pi}\frac{1}{n+1}\right)=\frac{6\ln2}{\pi}
\end{align}
\begin{align} & \sum_{0\leq n}\frac{(-1)^n}{(n+\frac{1}{2})^4\beta_n^3}\sum_{k=0}^{n}\beta_k^{3,-1}+\sum_{0< n}\frac{(-1)^{n-1}}{n^4\beta_n^3}\sum_{k=0}^{n-1}\beta_k^{3,-1}=4\pi\sum_{0< n}\frac{1}{n^2}\sum_{m=0}^{n-1}\frac{(-1)^m}{m+\frac{1}{2}} \end{align}
\begin{align}
\sum_{0\leq n}\frac{1}{(n+\frac{1}{2})^3\beta_n^2}\left(\sum_{k=0}^{n}\beta_k^{3,-1}\right)^2+\sum_{0< n}\frac{1}{n^3\beta_n^2}\left(\sum_{k=0}^{n-1}\beta_k^{3,-1}\right)^2=28\zeta(3)
\end{align}
\begin{align}
\sum_{0\leq n}\frac{(-1)^n}{(n+\frac{1}{2})^4\beta_n^3}\left(\sum_{k=0}^{n}\beta_k^{3,-1}\right)^2+\sum_{0< n}\frac{(-1)^{n-1}}{n^4\beta_n^3}\left(\sum_{k=0}^{n-1}\beta_k^{3,-1}\right)^2
&=\pi^4-4\pi\sum_{0< n}\frac{1}{n^2}\sum_{m=0}^{n-1}\frac{(-1)^m}{m+\frac{1}{2}}
\end{align}
\begin{align}
\sum_{0\leq n}\frac{1}{(n+\frac{1}{2})^5\beta_n^4}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^2+\sum_{0< n}\frac{1}{n^5\beta_n^4}\left(\sum_{k=0}^{n-1}\beta_k^{3,-1}\right)^2=16\pi^3 G-28\pi^2\zeta(3)
\end{align}
\begin{align} \sum_{0\leq n}\frac{1}{(2n+\frac{3}{2})(n+\frac{1}{2})^4\beta_n^4}\left(\sum_{k=0}^n\beta_k^{3,1}\right)\left(\sum_{k=0}^n\beta_k^2\right)=\frac{\pi^5}{12}A \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\sum_{k=0}^n\beta_k^{5,-1}=\frac{A^2}{3} \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\sum_{k=0}^n\beta_k^{5,-1}\sum_{m=0}^{k-1}\beta_{m+\frac{1}{2}}^{3,-1}\sum_{l=0}^m\beta_l^{3,-1}=\frac{A^2}{30} \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\left(\sum_{k=0}^n\beta_k^{5,-1}\right)\left(\sum_{k=0}^n\beta_k^{4,1}\right)=\frac{A^2}{2} \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\left(\sum_{k=0}^n\beta_k^{5,-1}\right)\left(\sum_{k=0}^n\beta_k^{3,-1}\right)&=\frac{2}{3}A^2 \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)\left(\sum_{k=0}^n\beta_k^{5,-1}\cdot \frac{1}{\pi}\left(2\ln2+\sum_{l=1}^{2k}\frac{1}{l}\right)\right)=\frac{3}{20}A^2 \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^3=\sum_{0\leq n}\left(2\beta_n^3-\beta_{n+\frac{1}{2}}^3\right) \end{align}
\begin{align} \sum_{0\leq n}\beta_{n}^{5,-1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^2=\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\left(\sum_{k=0}^n\beta_k^{4,1}\right)^2=\sum_{0\leq n}\left(\beta_n^3-\beta_{n+\frac{1}{2}}^3\right) \end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^4\left(\sum_{k=0}^n\beta_k^{4,1}\right)\left(\sum_{k=0}^n\beta_k^2\right)=\frac{1}{2}\sum_{0\leq n}\beta_n^4 \end{align}
\begin{align} \frac{1}{A}\sum_{0\leq n}\beta_n^{6,1}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-1}&=2\sum_{0\leq n}\beta_n^4-\sum_{0\leq n}\beta_n^{6,1}\\ A\sum_{0\leq n}\beta_n^{6,1}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-3}&=2\sum_{0\leq n}\beta_n^4+\sum_{0\leq n}\beta_n^{6,1} \end{align}
\begin{align}
\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{6,1}\sum_{k=0}^n\beta_k^{6,1}+2\sum_{0\leq n}\beta_{n+\frac{1}{2}}^4\sum_{k=0}^n\beta_k^4=\left(\sum_{0\leq n}\beta_n^4\right)^2
\end{align}
\begin{align}
&\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{6,1}\left(\sum_{k=0}^n\beta_k^{6,1}\right)\left(\sum_{k=0}^n\beta_k^{3,-1}\right)= \left(\sum_{0\leq n}\beta_n^4\right)^2 -\sum_{0\leq n}\beta_{n+\frac{1}{2}}^4\sum_{m=0}^n\beta_m^4\\
&=\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{6,1}\left(\sum_{k=0}^n\beta_k^{6,1}\sum_{k\leq m}\beta_{m+\frac{1}{2}}^{3,-1}\right)\left(\sum_{k=0}^n\beta_k^{4,1}\right)
\end{align}
\begin{align}
\sum_{0\leq n}\beta_n^4\left(\frac{8G}{\pi^2}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^2\right)\left(\frac{28\zeta(3)}{\pi^3}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^{4,1}\right)=\sum_{0\leq n}\beta_{n}^{6,1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^3+\frac{1}{2}\sum_{0\leq n}\beta_n^4
\end{align}
\begin{align}
\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{6,1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^3=\frac{3}{4}\sum_{0\leq n}\beta_n^{6,1}
\end{align}
\begin{align}
\sum_{0\leq n}\beta_{n+\frac{1}{2}}^4\left(\sum_{k=0}^n\beta_k^2\right)^3=\frac{1}{4}\sum_{0\leq n}\beta_n^{6,1}
\end{align}
\begin{align}
\sum_{0\leq n}\beta_n^{6,1}\left(\frac{8G}{\pi^2}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^2\right)^2=\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{6,1}\left(\sum_{k=0}^n\beta_k^2\right)^4
\end{align}
\begin{align} \sum_{0\leq n}\frac{(-1)^n\beta_n^5}{n+\frac{1}{2}}+\sum_{0< n}\frac{(-1)^n\beta_n^5}{n}=10\ln2-\frac{\pi}{2}\left(A\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-3}+\frac{1}{A}\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\right) \end{align}
\begin{align} \sum_{0\leq n}\frac{(-1)^n}{(n+\frac{1}{2})^6\beta_n^5} +\sum_{0< n}\frac{(-1)^{n-1}}{n^6\beta_n^5}=\frac{\pi^6}{2}\left(1-\left(\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-3}\right)\left(\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{5,-1}\right)\right) \end{align}
\begin{align} \sum_{0\leq n}\beta_n^{7,-1}\left(\frac{28\zeta(3)}{\pi^3}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^{4,1}\right)-\sum_{0\leq n}\beta_n^{7,-1}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}=\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{7,-1}\left(\sum_{k=0}^n\beta_k^{4,1}\right)^2 \end{align}
\begin{align} 4\sum_{0\leq n}\beta_n^{8,1}\left(\frac{28\zeta(3)}{\pi^3}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^{4,1}\right)-2\sum_{0\leq n}\beta_n^{8,1}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}&=\sum_{0\leq n}\beta_n^{8,1}\left(A\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-3}+\frac{1}{A}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-1}\right)\\ 3\sum_{0\leq n}\beta_n^{8,1}-\sum_{0\leq n}\beta_n^{8,1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^2 &=\sum_{0\leq n}\beta_n^{8,1}\left(A\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-3}-\frac{1}{A}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-1}\right) \end{align}
\begin{align} A^2\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{8,1}\left(\sum_{k=0}^n\beta_k^{5,-3}\right)^2-\frac{1}{A^2}\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{8,1}\left(\sum_{k=0}^n\beta_k^{5,-1}\right)^2=4\sum_{0\leq n}\beta_n^{8,1}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1} \end{align}
\begin{align}
3\sum_{0\leq n}\beta_n^{8,1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^2-2\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{8,1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^3=\sum_{0\leq n}\beta_n^{8,1}
\end{align}
\begin{align}
\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{8,1}\left(\sum_{k=0}^n\beta_{k}^{3,-1}\right)^2+\sum_{0\leq n}\beta_n^{8,1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-1}\right)\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-3}\right)&=\sum_{0\leq n}\beta_n^{8,1}\\
\sum_{0\leq n}\beta_{n}^{8,1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^2+\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{8,1}\left(\sum_{k=0}^n\beta_k^{5,-3}\right)\left(\sum_{k=0}^n\beta_k^{5,-1}\right)&=\sum_{0\leq n}\beta_n^{8,1}
\end{align}
\begin{align}
&\sum_{0\leq n}\beta_n^{9,-1}\left(\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{3,-1}\right)^3-\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{9,-1}\left(\sum_{k=0}^n\beta_k^{3,-1}\right)^3\\
&=\frac{1}{4}\sum_{0\leq n}\beta_{n+\frac{1}{2}}^{9,-1}\left(A\sum_{k=0}^n\beta_k^{5,-3}-\frac{1}{A}\sum_{k=0}^n\beta_k^{5,-1}\right)^2
\end{align}
\begin{align} \sum_{0\leq n}\beta_{n+\frac{1}{2}}^{10,1}\left(\sum_{k=0}^n\beta_k^{4,1}\right)^3+\sum_{0\leq n}\beta_n^{10,1}\left(A\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-3}+\frac{1}{A}\sum_{n\leq k}\beta_{k+\frac{1}{2}}^{5,-1}\right)=3\sum_{0\leq n}\beta_n^{10,1}\left(\frac{28\zeta(3)}{\pi^3}+\sum_{k=0}^{n-1}\beta_{k+\frac{1}{2}}^{4,1}\right) \end{align}
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