級数bot IIの級数まとめ

級数まとめ

級数$ \mathrm{bot II} $ の級数をまとめます。

$\displaystyle\frac1{\pi},\frac1{\pi^2}$の表示

$$\frac1{\pi}=\sum_{0\le n}\frac{(6n+1)\binom{2n}{n}^3}{2^{8n+2}}$$
$$\frac1{\pi}=\sum_{0\le n}\frac{(42n+5)\binom{2n}{n}^3}{2^{12n+4}}$$
$$\frac1{\pi^2}=\sum_{0\le n}\frac{(-1)^n\left(20n^2+8n+1\right)\binom{2n}{n}^5}{2^{12n+3}}$$
$$\frac1{\pi^2}=\sum_{0\le n}\frac{(-1)^n\left(820n^2+180n+13\right)\binom{2n}{n}^5}{2^{20n+7}}$$

$\displaystyle\zeta(2)$の表示

$$\zeta(2)=\sum_{0< m,n}\frac{(m-1)!(n-1)!}{(n+m)!}$$
$$\zeta(2)=3\sum_{0< n}\frac1{n^2\binom{2n}{n}}$$
$$\zeta(2)=\frac74\sum_{0< n}\frac1{n^2\binom{3n}{n}}+\sum_{0\le n}\frac1{(2n+1)^2\binom{3n+1}{n}}$$
$$\zeta(2)=\frac23\sum_{0< n}\frac{\binom{2n}{n}}{2^{2n}n}\sum_{k=0}^{n-1}\frac1{2k+1}$$
$$\zeta(2)=\frac53\sum_{0\le n}\frac{(-1)^n\binom{2n}{n}}{2^{4n}(2n+1)^2}$$
$$\zeta(2)=\sum_{0< n}\frac{21n-8}{n^3\binom{2n}{n}^3}$$
$$\zeta(2)=\frac1{3}\sum_{0< n}\frac{2^{4n}(3n-1)}{n^3\binom{2n}{n}^3}$$
$$\zeta(2)=\frac13\sum_{0\le n}\frac{2^{4n}(5n-1)}{n^3\binom{2n}{n}\binom{4n}{2n}^2}+\frac{16}{3}\sum_{0\le n}\frac{2^{4n}}{(2n+1)^2\binom{2n}{n}\binom{4n+2}{2n+1}}$$

$\displaystyle\zeta(3)$の表示

$$\zeta(3)=8\sum_{0< m< n}\frac{(-1)^n}{mn^2}$$
$$\zeta(3)=3\sum_{0< m< n}\frac1{mn^2\binom{2n}{n}}+2\sum_{0< n}\frac1{n^3\binom{2n}n}$$
$$\zeta(3)=\frac13\sum_{0< m,n}\frac{(m-1)!(n-1)!}{(n+m)!}\sum_{k=1}^{n+m}\frac1{k}$$
$$\zeta(3)=\frac23\sum_{0< m< n}\frac{\binom{2n}{n}}{2^{2n}m^2n}$$
$$\zeta(3)=\frac17\sum_{0< m< n}\frac{2^{2n}}{mn^2\binom{2n}{n}}$$
$$\zeta(3)=\frac52\sum_{0< n}\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$
$$\zeta(3)=\sum_{0< n}\frac1{n^2\binom{2n}{n}}\sum_{k=1}^{2n}\frac1{k}+\frac23\sum_{0< n}\frac1{n^3\binom{2n}{n}}$$
$$\zeta(3)=\frac{2\pi}7\sum_{0\le m\le n}\frac{\binom{2m}{m}^2}{2^{4m}(2n+1)^2}$$
$$\zeta(3)=8\sum_{0< m< n< r\le 2n}\frac{\binom{2n}{n}}{2^{2n}mnr}$$
$$\zeta(3)=9\sum_{0< m< n}\frac{\binom{2m}{m}}{mn^2\binom{2n}{n}}$$
$$\zeta(3)=\frac{15}{16}\sum_{0< n}\frac{(-1)^{n-1}}{n^2\binom{2n}{n}}\sum_{k=1}^{2n}\frac1{k}+\frac58\sum_{0\le n}\frac{(-1)^n}{(2n+1)^2\binom{2n}{n}}\sum_{k=1}^{2n+1}\frac1{k}$$
$$\zeta(3)=\frac54\sum_{0< n}\frac{(-1)^n}{n^3\binom{2n}{n}\binom{3m}{n}}+\sum_{0\le n}\frac{(-1)^n}{(2n+1)^3\binom{2n}{n}\binom{3n+1}{n}}$$
$$\zeta(3)=\frac1{28}\sum_{0< n}\frac{(-1)^{n-1}2^{8n}\left(10n^2-6n+1\right)}{n^5\binom{2n}{n}^5}$$
$$\zeta(3)=\frac12\sum_{0< n}\frac{(-1)^{n-1}\left(205n^2-160n+32\right)}{n^5\binom{2n}{n}^5}$$
$$\zeta(3)=\frac{11}{4}\sum_{0< n}\frac1{n^3\binom{2n}{n}^2}+\frac12\sum_{0\le n}\frac1{(2n+1)^3\binom{2n}{n}^2}$$

$\displaystyle\zeta(4)$の表示

$$\zeta(4)=4\sum_{0< m< n}\frac1{mn^3}$$
$$\zeta(4)=\frac45\sum_{0< m< n< r}\frac{1}{mnr(r-m)}$$
$$\zeta(4)=\frac87\sum_{0< m< n< r}\frac1{mnr(m+r)}$$
$$\zeta(4)=\frac{8}{17}\sum_{0< m,n}\frac{(m-1)!(n-1)!}{(m+n)!}\sum_{k=1}^{m+n}\frac1{k^2}$$
$$\zeta(4)=\frac{36}{17}\sum_{0< n}\frac1{n^4\binom{2n}{n}}$$
$$\zeta(4)=\sum_{0< n}\frac{(-1)^n}{n^4\binom{2n}{n}}-\frac{10}{3}\sum_{0< m\le n}\frac{(-1)^n}{mn^3\binom{2n}{n}}$$
$$\zeta(4)=4\sum_{0< m< n}\frac{\binom{2m}{m}}{m^2n^2\binom{2n}{n}}+12\sum_{0< m< n< r}\frac{\binom{2n}{n}}{mnr^2\binom{2r}{r}}$$
$$\zeta(4)=\frac32\sum_{0< n}\frac{(-1)^n}{n^4\binom{2n}{n}\binom{3n}{n}}+\frac43\sum_{0\le n}\frac{(-1)^n}{(2n+1)^4\binom{2n}{n}\binom{3n+1}{n}}$$

$\zeta(5)$の表示

$$\zeta(5)=\sum_{0< m< n< r}\frac1{m^2n(r-m)^2}$$
$$\zeta(5)=\frac12\sum_{0< m< n< r< s}\frac{1}{mnrs(s-r+n-m)}$$
$$\zeta(5)=2\sum_{0< n}\frac{(-1)^{n-1}}{n^5\binom{2n}{n}}+\frac52\sum_{0< m< n}\frac{(-1)^n}{m^2n^3\binom{2n}n}$$

$\displaystyle\zeta(6)$の表示

$$\zeta(6)=48\sum_{0< m< n< r}\frac1{n^3r^2(r-m)}$$

$\displaystyle\beta(2)$の表示

$$\beta(2)=\frac12\sum_{0\le n}\frac{2^{2n}}{(2n+1)^2\binom{2n}{n}}$$
$$\beta(2)=\frac12\sum_{0\le m\le n}\frac{2^n}{(2m+1)(2n+1)\binom{2n}{n}}$$
$$\beta(2)=\frac12\sum_{0< m< n}\frac{2^{m+n}}{mn\binom{2n}{n}}$$
$$\beta(2)=\frac{\pi}{4}\sum_{0\le n}\frac{\binom{2n}{n}^2}{2^{4n}(2n+1)}$$
$$\beta(2)=\frac{3\pi}{4}\sum_{0\le m< n}\frac{2^{4m-8n}(6n+1)\binom{2n}{n}^3}{(2m+1)^2\binom{2m}{m}}$$

二項係数を含む級数

$$\sum_{0< n}\frac{\binom{2n}{n}}{2^{2n}n^2}=\frac{\pi^2}{6}-2\log^22$$
$$\sum_{0< n}\frac{2^{2n}}{n^3\binom{2n}{n}}=\pi^2\log2-\frac72\zeta(3)$$
$$\sum_{0\le n}\frac1{(2n+1)^2\binom{2n}{n}}=\frac83\beta(2)-\frac{\pi}3\log\left(2+\sqrt{3}\right)$$
$$\sum_{0\le n}\frac{(-1)^n}{(2n+1)^2\binom{2n}{n}}=\frac{\pi^2}{6}-3\log^2\left(\frac{1+\sqrt{5}}{2}\right)$$
$$\sum_{0\le n}\frac{\binom{2n}{n}}{2^{4n}(2n+1)^2}=\frac{\sqrt{3}}2\left(1+\frac1{2^2}-\frac1{4^2}-\frac1{5^2}+\frac1{7^2}+\frac1{8^2}-\cdots\right)$$
$$\sum_{0\le n}\frac{\binom{2n}{n}}{2^{4n}(2n+1)^3}=\frac{7\pi^3}{216}$$
$$\sum_{0\le m\le n}\frac{\binom{2n}{n}}{2^{3n}(2m+1)(2n+1)}=\frac{3\pi}{4\sqrt{2}}\log2$$
$$\sum_{0< m< n}\frac{2^{2n-2m}\binom{2m}{m}}{mn^2\binom{2n}{n}}=\pi^2\log2-\frac72\zeta(3)$$
$$\pi\sum_{0< n}\frac{\binom{2n}{n}^2}{2^{4n}n}=4\pi\log2-8\beta(2)$$
$$\sum_{0< n}\left(\frac{2^{4n}}{n^2\binom{2n}{n}^2}-\frac{\pi}{n}\right)=4\pi\log2-8\beta(2)$$
$$\sum_{0< n}\frac{2^{4n}}{n^3\binom{2n}{n}^2}=8\pi\log2-14\zeta(3)$$
$$\sum_{0< n}\frac{3^{3n}}{n^3\binom{2n}{n}\binom{3n}{n}}=9\sqrt3\pi\left(1-\frac1{2^2}+\frac1{4^2}-\frac1{5^2}+\cdots\right)-26\zeta(3)$$
$$\sum_{0< n}\frac{2^{2n}\binom{3n}{n}}{3^{3n}n}=3\log\frac32$$
$$\sum_{0< n}\frac{\binom{3n}{n}\binom{6n}{3n}}{2^{2n}3^{3n}n\binom{2n}{n}}=\log\frac{\left(3-\sqrt{3}\right)^6}{2}$$
$$\sum_{0\le n}\frac{2^{2n}\binom{3n+1}{n}}{3^{3n}(2n+1)}=\frac{3\sqrt{3}}{4}\log\left(2+\sqrt{3}\right)$$
$$\sum_{0< n}\frac{1}{2^nn^2\binom{3n}{n}}=\frac{\pi^2}{24}-\frac12\log^22$$
$$\sum_{0< n}\frac{3^{3n}}{2^{2n}n^2\binom{3n}{n}}=\frac{2}{3}\pi^2-2\log^23$$
$$\sum_{0< n}\frac{2^{8n}}{3^{3n}n^2\binom{4n}{n}}=\frac{3}{4}\left(\pi^2-2\log^23+\arctan^2\frac{4\sqrt2}{7}\right)$$
$$\sum_{0< n}\frac{\binom{4n}{2n}}{2^{2n}n^2\binom{2n}{n}}=\frac{\pi^2}{6}-\log^2\left(\frac{1+\sqrt{2}}{2}\right)+2\mathrm{Li}_2\left(\frac{1-\sqrt{2}}{2}\right)$$
$$\sum_{0\le n}\frac{\binom{2n}{n}^3}{2^{6n}}=\frac{\pi}{\Gamma\left(\frac34\right)^4}$$
$$\sum_{0< n}\frac{2^{6n}}{n^3\binom{2n}{n}^3}=8\pi\sum_{0\le 2n< m}\frac{\binom{2n}{n}^2}{2^{4n}(4n+1)}\frac{(-1)^{m-1}}{m}$$
$$\sum_{0\le n}\frac{(-1)^n\binom{2n}{n}}{2^{2n}(4n+1)^3}=\frac{\Gamma\left(\frac14\right)^4}{16\sqrt{\pi}}\left(\beta(2)+\frac{\pi^2}{8}\right)$$

$$\sum_{0< m,n}\frac1{mn}\left(\frac{m!n!}{(m+n)!}\right)^2=\frac{\pi^2}6-\left(1+\frac1{2^2}-\frac1{4^2}-\frac1{5^2}+\frac1{7^2}+\frac1{8^2}-\cdots\right)$$
$$\sum_{0< m,n,r}\frac{(m-1)!(n-1)!(r-1)!}{(m+n+r)!}=\frac{13}{4}\zeta(3)-\frac{\pi^2}2\log2$$
$$?=\sum_{0\le n}\frac{2^{2n}}{(4n+1)^2\binom{2n}{n}}$$

テータ関数の特殊値

$$\sum_{n\in\mathbb{Z}}e^{-\pi n^2}=\frac{\sqrt[4]{\pi}}{\Gamma\left(\frac34\right)}$$

$2$重級数

$$\sum_{n,m\in\mathbb{Z}}\frac1{\cosh\pi n+i\sinh \pi n}=\frac{\pi}{\Gamma\left(\frac34\right)^4}$$

$\displaystyle\mathrm{Eisenstein}$級数の特殊値

$$\sum_{0< n}\frac{n^3}{e^{2\pi n}-1}=\frac{\pi^2}{320\Gamma\left(\frac34\right)^8}-\frac1{240}$$
$$\sum_{0< n}\frac{n^5}{e^{2\pi n}-1}=\frac{1}{504}$$

$\displaystyle\sinh$を含む級数

$$\sum_{0< n}\frac{(-1)^{n-1}n}{\sinh\pi n}=\frac1{4\pi}$$

$\displaystyle\sin$を含む級数

$$\sum_{k=0}^{n-1}\frac1{\sin^2\left(x+\frac{\pi k}{n}\right)}=\frac{n^2}{\sin^2nx}$$

$q$級数

$$\frac{(q;q)_{\infty}}{(-q;q)_{\infty}}=\sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}$$
$$\frac1{(q;q)_{\infty}}=\sum_{0\le n}\frac{q^{n^2}}{(q;q)^2_n}$$
$$\frac1{\left(q,q^4;q^5\right)_{\infty}}=\sum_{0\le n}\frac{q^{n^2}}{(q;q)_n}$$
$$\frac1{\left(q^2,q^3;q^5\right)_{\infty}}=\sum_{0\le n}\frac{q^{n(n+1)}}{(q;q)_n}$$
$$\frac{\left(q^2;q^2\right)_\infty}{\left(q;q^2\right)_\infty}=\sum_{0< n}q^{\binom{n}{2}}$$
$$\left(q;q\right)^3_{\infty}=\sum_{0< n}(-1)^{n-1}(2n-1)q^{\binom{n}{2}}$$
$$\sum_{0< m< n}\frac{(-1)^nq^{\binom{n}{2}}}{\left(1-q^m\right)(q;q)_n}=\sum_{0< n}\frac{nq^n}{1-q^n}$$
$$\sum_{0< m< n}\frac{q^n}{\left(1-q^m\right)\left(1-q^m\right)^2}=\sum_{0< n}\frac{q^{2n}}{\left(1-q^n\right)^3}$$

変数付きゼータ関数

$$\sum_{0< n}\frac{n!}{n^2(x)_n}=\sum_{0\le n}\frac{1}{(n+x)^2}$$
$$\sum_{0\le m< n}\frac{(x)_m}{m!}\frac{1}{(m+x)(n+x)^2}\frac{n!}{(x)_n}=\sum_{0\le n}\frac{1}{(n+x)^3}$$
$$\sum_{0\le n}\frac{(2x)_n}{n!}\frac{(-1)^n}{(n+x)^2}=2^{-2x}\sum_{0\le n}\frac{1}{(n+x)^2}\frac{\left(\frac12+x\right)_n}{(x)_n}$$
$$2\sum_{0\le n}\frac{(2x)_n}{n!}\frac{(-1)^n}{(n+x)^3}=\frac{\Gamma(x)^2}{\Gamma(2x)}\sum_{0\le n}\frac1{(n+x)^2}$$
$$2\sum_{0\le n}\frac{(-1)^n}{n+x}\frac{n!}{(2x)_n}=\sum_{0\le n}\frac{(x)_n}{(n+x)(2x)_n}$$

有限級数

$$\sum_{n=1}^{p-1}\frac1{n}=0\quad\left(\mathrm{mod}\ p^2\right)$$
$$\sum_{n=1}^{\frac{p-1}{2}}\frac1{n}=-\frac{2\left(2^{p-1}-1\right)}{p}\quad(\mathrm{mod}\ p)$$
$$\sum_{n=1}^{p-1}\frac{F_n}{n}=0\quad(\mathrm{mod}\ p)$$
$$\sum_{n=1}^{p-1}\frac{\binom{2n}{n}}{n}=0\quad(\mathrm{mod}\ p)$$
$$\frac{F_{p-\left(\frac{5}{p}\right)}}{p}=\frac15\sum_{n=1}^{p-1}(-1)^{n-1}\frac{\binom{2n}{n}}{n}\quad(\mathrm{mod}\ p)$$

$\displaystyle\mathrm{Hasse}$の級数表示

$$\zeta(s)=\frac1{s-1}\sum_{0< n}\frac{1}{n}\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^{s-1}}\binom{n-1}{k-1}$$

$\gamma$の表示

$$\gamma=\sum_{1< n}\frac{(-1)^n}{n}\zeta(n)$$
$$\gamma=\sum_{0< n}\frac1n\sum_{k=1}^{n}{(-1)^k}\binom{n-1}{k-1}\log k$$

多重ゼータ値

$$\sum_{0< n_1<\cdots< n_r}\frac{1}{n_1^2\cdots n_r^2}=\frac{\pi^{2r}}{(2r+1)!}$$

二項係数を含むべき級数

$$\sum_{0< n}\frac{\binom{2n}{n}}{2^{2n}n}x^n=2\log\left(\frac2{1+\sqrt{1-x}}\right)$$

$\displaystyle\arcsin$の累乗のべき級数展開

$$\frac{(2\arcsin x)^{2r}}{(2r)!}=\sum_{0< n_1<\cdots< n_r}\frac{2^{2n_r}}{n_1^2\cdots n_r^2\binom{2n_r}{n_r}}x^{2n_r}$$
$$\frac{(2\arcsin x)^{2r-1}}{(2r-1)!}=\sum_{0\le n_1<\cdots< n_r}\frac{\binom{2n_r}{n_r}}{{2^{2n_r}\left(n_1+\frac12\right)^2\cdots}\left(n_{r-1}+\frac12\right)^2\left(n_r+\frac12\right)}x^{2n_r+1}$$

恒等式

$$\sum_{0\le k}\frac{(-1)^{k}}{2^{2k}}\binom{n-k-1}{k}x^k=\frac1{\sqrt{1-x}}\left(\left(\frac{1+\sqrt{1-x}}{2}\right)^n-\left(\frac{1-\sqrt{1-x}}{2}\right)^n\right)$$
$$\left(\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n+k}{k}x^k\right)^2=\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n+k}{k}\binom{2k}{k}x^k(1-x)^k$$