ハイパー調和数の定義
ハイパー調和数の定義
ハイパー調和数\(H_{n}^{\left(r\right)}\)を次で定義する。
\[ H_{n}^{\left(r\right)}:=\begin{cases} \frac{1}{n} & r=0\\ \sum_{k=1}^{n}H_{k}^{\left(r-1\right)} & r\in\mathbb{N} \end{cases} \]
ハイパー調和数\(H_{n}^{\left(r\right)}\)を次で定義する。
\[ H_{n}^{\left(r\right)}:=\begin{cases} \frac{1}{n} & r=0\\ \sum_{k=1}^{n}H_{k}^{\left(r-1\right)} & r\in\mathbb{N} \end{cases} \]
\begin{align*}
H_{3}^{\left(2\right)} & =\sum_{k=1}^{3}H_{k}^{\left(1\right)}\\
& =\sum_{k=1}^{3}H_{k}\\
& =H_{1}+H_{2}+H_{3}\\
& =1+1+\frac{1}{2}+1+\frac{1}{2}+\frac{1}{3}\\
& =3+\frac{2}{2}+\frac{1}{3}\\
& =\frac{13}{3}
\end{align*}
ハイパー調和数一覧
\[ \begin{array}{|c|c|c|c|c|c|c|} \hline n & H_{n}^{\left(0\right)} & H_{n}^{\left(1\right)} & H_{n}^{\left(2\right)} & H_{n}^{\left(3\right)} & H_{n}^{\left(4\right)} & H_{n}^{\left(5\right)}\\ \hline 0 & \infty & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 2 & \frac{1}{2} & \frac{3}{2} & \frac{5}{2} & \frac{7}{2} & \frac{9}{2} & \frac{11}{2}\\ \hline 3 & \frac{1}{3} & \frac{11}{6} & \frac{13}{3} & \frac{47}{6} & \frac{37}{3} & \frac{107}{6}\\ \hline 4 & \frac{1}{4} & \frac{25}{12} & \frac{77}{12} & \frac{57}{4} & \frac{319}{12} & \frac{533}{12}\\ \hline 5 & \frac{1}{5} & \frac{137}{60} & \frac{87}{10} & \frac{459}{20} & \frac{743}{15} & \frac{1,879}{20}\\ \hline 6 & \frac{1}{6} & \frac{49}{20} & \frac{223}{20} & \frac{341}{10} & \frac{2,509}{30} & \frac{2,131}{12}\\ \hline 7 & \frac{1}{7} & \frac{363}{140} & \frac{481}{35} & \frac{3,349}{70} & \frac{2,761}{21} & \frac{25,961}{84}\\ \hline 8 & \frac{1}{8} & \frac{761}{280} & \frac{4,609}{280} & \frac{3,601}{56} & \frac{32,891}{168} & \frac{28,271}{56}\\ \hline 9 & \frac{1}{9} & \frac{7,129}{2,520} & \frac{4,861}{252} & \frac{42,131}{504} & \frac{35,201}{126} & \frac{395,243}{504}\\ \hline 10 & \frac{1}{10} & \frac{7,381}{2,520} & \frac{55,991}{2,520} & \frac{44,441}{420} & \frac{485,333}{1,260} & \frac{420,983}{360}\\ \hline 11 & \frac{1}{11} & \frac{83,711}{27,720} & \frac{58,301}{2,310} & \frac{605,453}{4,620} & \frac{511,073}{990} & \frac{445,007}{264}\\ \hline 12 & \frac{1}{12} & \frac{86,021}{27,720} & \frac{785,633}{27,720} & \frac{631,193}{3,960} & \frac{535,097}{792} & \frac{935,059}{396}\\ \hline 13 & \frac{1}{13} & \frac{1,145,993}{360,360} & \frac{811,373}{25,740} & \frac{655,217}{3,432} & \frac{1,115,239}{1,287} & \frac{16,616,723}{5,148}\\ \hline 14 & \frac{1}{14} & \frac{1,171,733}{360,360} & \frac{835,397}{24,024} & \frac{1,355,479}{6,006} & \frac{19,679,783}{18,018} & \frac{17,297,403}{4,004}\\ \hline 15 & \frac{1}{15} & \frac{1,195,757}{360,360} & \frac{1,715,839}{45,045} & \frac{23,763,863}{90,090} & \frac{6,786,821}{5,005} & \frac{113,634,299}{20,020}\\ \hline 16 & \frac{1}{16} & \frac{2,436,559}{720,720} & \frac{29,889,983}{720,720} & \frac{24,444,543}{80,080} & \frac{133,033,679}{80,080} & \frac{117,514,175}{16,016}\\ \hline 17 & \frac{1}{17} & \frac{42,142,223}{12,252,240} & \frac{30,570,663}{680,680} & \frac{476,698,557}{1,361,360} & \frac{136,913,555}{68,068} & \frac{363,627,885}{38,896}\\ \hline 18 & \frac{1}{18} & \frac{14,274,301}{4,084,080} & \frac{197,698,279}{4,084,080} & \frac{162,779,395}{408,408} & \frac{140,608,675}{58,344} & \frac{124,736,455}{10,608}\\ \hline 19 & \frac{1}{19} & \frac{275,295,799}{77,597,520} & \frac{201,578,155}{3,879,876} & \frac{166,474,515}{369,512} & \frac{144,135,835}{50,388} & \frac{2,946,535,985}{201,552}\\ \hline 20 & \frac{1}{20} & \frac{55,835,135}{15,519,504} & \frac{41,054,655}{739,024} & \frac{34,000,335}{67,184} & \frac{678,544,345}{201,552} & \frac{604,180,055}{33,592}\\ \hline 21 & \frac{1}{21} & \frac{18,858,053}{5,173,168} & \frac{13,920,029}{235,144} & \frac{265,842,403}{470,288} & \frac{693,417,203}{176,358} & \frac{15,461,449,967}{705,432}\\ \hline 22 & \frac{1}{22} & \frac{19,093,197}{5,173,168} & \frac{325,333,835}{5,173,168} & \frac{812,400,067}{1,293,292} & \frac{17,692,378,667}{3,879,876} & \frac{15,804,669,767}{596,904}\\ \hline 23 & \frac{1}{23} & \frac{444,316,699}{118,982,864} & \frac{990,874,363}{14,872,858} & \frac{20,666,950,267}{29,745,716} & \frac{18,035,598,467}{3,432,198} & \frac{145,216,599,503}{4,576,264}\\ \hline 24 & \frac{1}{24} & \frac{1,347,822,955}{356,948,592} & \frac{25,128,807,667}{356,948,592} & \frac{21,010,170,067}{27,457,584} & \frac{165,294,957,803}{27,457,584} & \frac{148,084,936,403}{3,922,512}\\ \hline 25 & \frac{1}{25} & \frac{34,052,522,467}{8,923,714,800} & \frac{25,472,027,467}{343,219,800} & \frac{192,066,102,203}{228,813,200} & \frac{168,163,294,703}{24,515,700} & \frac{4,374,776,588,887}{98,062,800}\\ \hline 26 & \frac{1}{26} & \frac{34,395,742,267}{8,923,714,800} & \frac{232,222,818,803}{2,974,571,600} & \frac{194,934,439,103}{212,469,400} & \frac{4,957,048,979,587}{637,408,200} & \frac{4,452,412,907,647}{84,987,760}\\ \hline 27 & \frac{1}{27} & \frac{312,536,252,003}{80,313,433,200} & \frac{235,091,155,703}{2,868,336,900} & \frac{5,733,412,167,187}{5,736,673,800} & \frac{5,034,685,298,347}{573,667,380} & \frac{140,353,889,699,857}{2,294,669,520}\\ \hline 28 & \frac{1}{28} & \frac{315,404,588,903}{80,313,433,200} & \frac{6,897,956,948,587}{80,313,433,200} & \frac{5,811,048,485,947}{5,354,228,880} & \frac{158,404,333,811,557}{16,062,686,640} & \frac{285,220,390,427,639}{4,015,671,660}\\ \hline 29 & \frac{1}{29} & \frac{9,227,046,511,387}{2,329,089,562,800} & \frac{6,975,593,267,347}{77,636,318,760} & \frac{182,471,592,627,157}{155,272,637,520} & \frac{321,321,278,651,039}{29,113,619,535} & \frac{289,596,255,666,839}{3,528,923,580}\\ \hline 30 & \frac{1}{30} & \frac{9,304,682,830,147}{2,329,089,562,800} & \frac{218,572,480,850,557}{2,329,089,562,800} & \frac{369,455,796,282,239}{291,136,195,350} & \frac{325,697,143,890,239}{26,466,926,850} & \frac{293,843,418,987,239}{3,113,756,100} \\\hline \end{array} \]
\[ \begin{array}{|c|c|c|c|c|c|c|} \hline n & H_{n}^{\left(0\right)} & H_{n}^{\left(1\right)} & H_{n}^{\left(2\right)} & H_{n}^{\left(3\right)} & H_{n}^{\left(4\right)} & H_{n}^{\left(5\right)}\\ \hline 0 & \infty & 0 & 0 & 0 & 0 & 0\\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1\\ \hline 2 & \frac{1}{2} & \frac{3}{2} & \frac{5}{2} & \frac{7}{2} & \frac{9}{2} & \frac{11}{2}\\ \hline 3 & \frac{1}{3} & \frac{11}{6} & \frac{13}{3} & \frac{47}{6} & \frac{37}{3} & \frac{107}{6}\\ \hline 4 & \frac{1}{4} & \frac{25}{12} & \frac{77}{12} & \frac{57}{4} & \frac{319}{12} & \frac{533}{12}\\ \hline 5 & \frac{1}{5} & \frac{137}{60} & \frac{87}{10} & \frac{459}{20} & \frac{743}{15} & \frac{1,879}{20}\\ \hline 6 & \frac{1}{6} & \frac{49}{20} & \frac{223}{20} & \frac{341}{10} & \frac{2,509}{30} & \frac{2,131}{12}\\ \hline 7 & \frac{1}{7} & \frac{363}{140} & \frac{481}{35} & \frac{3,349}{70} & \frac{2,761}{21} & \frac{25,961}{84}\\ \hline 8 & \frac{1}{8} & \frac{761}{280} & \frac{4,609}{280} & \frac{3,601}{56} & \frac{32,891}{168} & \frac{28,271}{56}\\ \hline 9 & \frac{1}{9} & \frac{7,129}{2,520} & \frac{4,861}{252} & \frac{42,131}{504} & \frac{35,201}{126} & \frac{395,243}{504}\\ \hline 10 & \frac{1}{10} & \frac{7,381}{2,520} & \frac{55,991}{2,520} & \frac{44,441}{420} & \frac{485,333}{1,260} & \frac{420,983}{360}\\ \hline 11 & \frac{1}{11} & \frac{83,711}{27,720} & \frac{58,301}{2,310} & \frac{605,453}{4,620} & \frac{511,073}{990} & \frac{445,007}{264}\\ \hline 12 & \frac{1}{12} & \frac{86,021}{27,720} & \frac{785,633}{27,720} & \frac{631,193}{3,960} & \frac{535,097}{792} & \frac{935,059}{396}\\ \hline 13 & \frac{1}{13} & \frac{1,145,993}{360,360} & \frac{811,373}{25,740} & \frac{655,217}{3,432} & \frac{1,115,239}{1,287} & \frac{16,616,723}{5,148}\\ \hline 14 & \frac{1}{14} & \frac{1,171,733}{360,360} & \frac{835,397}{24,024} & \frac{1,355,479}{6,006} & \frac{19,679,783}{18,018} & \frac{17,297,403}{4,004}\\ \hline 15 & \frac{1}{15} & \frac{1,195,757}{360,360} & \frac{1,715,839}{45,045} & \frac{23,763,863}{90,090} & \frac{6,786,821}{5,005} & \frac{113,634,299}{20,020}\\ \hline 16 & \frac{1}{16} & \frac{2,436,559}{720,720} & \frac{29,889,983}{720,720} & \frac{24,444,543}{80,080} & \frac{133,033,679}{80,080} & \frac{117,514,175}{16,016}\\ \hline 17 & \frac{1}{17} & \frac{42,142,223}{12,252,240} & \frac{30,570,663}{680,680} & \frac{476,698,557}{1,361,360} & \frac{136,913,555}{68,068} & \frac{363,627,885}{38,896}\\ \hline 18 & \frac{1}{18} & \frac{14,274,301}{4,084,080} & \frac{197,698,279}{4,084,080} & \frac{162,779,395}{408,408} & \frac{140,608,675}{58,344} & \frac{124,736,455}{10,608}\\ \hline 19 & \frac{1}{19} & \frac{275,295,799}{77,597,520} & \frac{201,578,155}{3,879,876} & \frac{166,474,515}{369,512} & \frac{144,135,835}{50,388} & \frac{2,946,535,985}{201,552}\\ \hline 20 & \frac{1}{20} & \frac{55,835,135}{15,519,504} & \frac{41,054,655}{739,024} & \frac{34,000,335}{67,184} & \frac{678,544,345}{201,552} & \frac{604,180,055}{33,592}\\ \hline 21 & \frac{1}{21} & \frac{18,858,053}{5,173,168} & \frac{13,920,029}{235,144} & \frac{265,842,403}{470,288} & \frac{693,417,203}{176,358} & \frac{15,461,449,967}{705,432}\\ \hline 22 & \frac{1}{22} & \frac{19,093,197}{5,173,168} & \frac{325,333,835}{5,173,168} & \frac{812,400,067}{1,293,292} & \frac{17,692,378,667}{3,879,876} & \frac{15,804,669,767}{596,904}\\ \hline 23 & \frac{1}{23} & \frac{444,316,699}{118,982,864} & \frac{990,874,363}{14,872,858} & \frac{20,666,950,267}{29,745,716} & \frac{18,035,598,467}{3,432,198} & \frac{145,216,599,503}{4,576,264}\\ \hline 24 & \frac{1}{24} & \frac{1,347,822,955}{356,948,592} & \frac{25,128,807,667}{356,948,592} & \frac{21,010,170,067}{27,457,584} & \frac{165,294,957,803}{27,457,584} & \frac{148,084,936,403}{3,922,512}\\ \hline 25 & \frac{1}{25} & \frac{34,052,522,467}{8,923,714,800} & \frac{25,472,027,467}{343,219,800} & \frac{192,066,102,203}{228,813,200} & \frac{168,163,294,703}{24,515,700} & \frac{4,374,776,588,887}{98,062,800}\\ \hline 26 & \frac{1}{26} & \frac{34,395,742,267}{8,923,714,800} & \frac{232,222,818,803}{2,974,571,600} & \frac{194,934,439,103}{212,469,400} & \frac{4,957,048,979,587}{637,408,200} & \frac{4,452,412,907,647}{84,987,760}\\ \hline 27 & \frac{1}{27} & \frac{312,536,252,003}{80,313,433,200} & \frac{235,091,155,703}{2,868,336,900} & \frac{5,733,412,167,187}{5,736,673,800} & \frac{5,034,685,298,347}{573,667,380} & \frac{140,353,889,699,857}{2,294,669,520}\\ \hline 28 & \frac{1}{28} & \frac{315,404,588,903}{80,313,433,200} & \frac{6,897,956,948,587}{80,313,433,200} & \frac{5,811,048,485,947}{5,354,228,880} & \frac{158,404,333,811,557}{16,062,686,640} & \frac{285,220,390,427,639}{4,015,671,660}\\ \hline 29 & \frac{1}{29} & \frac{9,227,046,511,387}{2,329,089,562,800} & \frac{6,975,593,267,347}{77,636,318,760} & \frac{182,471,592,627,157}{155,272,637,520} & \frac{321,321,278,651,039}{29,113,619,535} & \frac{289,596,255,666,839}{3,528,923,580}\\ \hline 30 & \frac{1}{30} & \frac{9,304,682,830,147}{2,329,089,562,800} & \frac{218,572,480,850,557}{2,329,089,562,800} & \frac{369,455,796,282,239}{291,136,195,350} & \frac{325,697,143,890,239}{26,466,926,850} & \frac{293,843,418,987,239}{3,113,756,100} \\\hline \end{array} \]
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調和数の相反公式
\[
H_{1-z}-H_{z}=\pi\tan^{-1}\left(\pi z\right)+\frac{1}{1-z}-\frac{1}{z}
\]
調和数・一般化調和数の積分
\[
\int H_{z}dz=\log\Gamma\left(z+1\right)+\gamma z+C
\]
調和数の特殊値
\[
H_{\frac{1}{2}}=2-2\log2
\]
調和数・一般化調和数を含む総和
\[
\sum_{k=1}^{n}H_{k,m}=\left(n+1\right)H_{n,m}-H_{n,m-1}
\]