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モリアーティ教授と二項定理
ホームズの宿敵「犯罪界のナポレオン」は
いかほどの数学者だったのか?
なんでもLT会 2015 Feb. 28@Lab-Cafe
原 啓介 (Mynd, Inc.)
Meet Prof. J. Moriarty (?-1891)
……生まれつき数学では抜群の才能も持って
いる。若干二十一歳で、<二項定理>に関す
る専門書を書いていて、これはその後ひとつ
の潮流として、ヨーロッパじゅうでもてはや
されてきた。その力もあって、ある小規模な
大学の数学教授職を獲得し、どこから見ても…
「回想のシャーロック・ホームズ」(A.C.ドイル著/
深町眞理子訳/創元推理文庫)所収、「最後の事件」より
<二項定理>に関する専門書: A Treatise on the Binomial Theorem
(x + y)2
= x2
+ 2xy + y2
(x + y)3
= x3
+ 3x2
y + 3xy2
+ y3
二項定理(Omar Khayyám, 1048-1131)
(x + y)n
=
nX
k=0
✓
n
k
◆
xn k
yk
=
nX
k=0
n!
k!(n k)!
xn k
yk
子どもの頃の私の疑問
「悪の天才モリアーティ教授たるものの
数学論文が、こんな簡単そうな
公式の話なのだろうか?」
n pX
k=0
✓
2n + 1
2p + 2k + 1
◆ ✓
p + k
k
◆
=
✓
2n p
p
◆
22n 2p
n pX
k=0
✓
2n
2p + 2k
◆ ✓
p + k
k
◆
=
n
2n p
✓
2n p
p
◆
22n 2p
モリアーティの公式? (H.T.Davis 1962による)
H.T.Davis: The summation of Series , Principe Press of Trinity
University, San Antonio, Texas, (1962)
“We shall call these the identities of James Moriarty, since we do not know
any other source from which such ingenious formulas could have come”.
(in a footnote of H.T.Davis)
• 「モリアーティの公式」は見かけは複雑だが、数学としての
深みがなさげで、天才的数学者の仕事としては疑わしい。
• もっともらしい可能性としては、代数的な拡張、組合せ論へ
の応用。一見遠そうだが、解析学的な性質の研究?(無限小
解析と二項定理には深い関係がある)
• 聖典「恐怖の谷」によれば、「小惑星の力学」( The
Dynamics of an Asteroid )という専門書も執筆している。
• Newton と同じく、天文学的動機を持つ力学系に関係した
解析学的な結果だったのではないか。
一般二項定理 (I.Newton, 1665?)
(x + y)↵
=
1X
k=0
↵!
k!(↵ k)!
x↵ k
yk
ただし、階乗はガンマ関数に拡張して
z! = (z + 1) と解釈する。
実数 |x| > |y| と実数(後に複素数に拡張) ↵
に対して、
例えば、こんな結果だったのでは……
↵
nX
j=0
(↵n)!
(↵j)!(↵(n j))!
x↵j
y↵(n j)
 (x + y)↵n
(See, Bulletin of the London Math. Sci.,
Vol.42, pp.467̶477, (2010),
or arXiv:1001.1775)
Theorem (K.Hara and M.Hino, 2010)
For any x, y > 0, 0 < α < 1, and a natural
number n, we have

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モリアーティ教授と二項定理