查看“︁普羅海特-蘇-摩爾斯常數”︁的源代码

{{Infobox number
| name=普羅海特-蘇-摩爾斯常數
| nav=no
| is integer = no
| number=<math>\tau</math>
| symbol=<math>\tau</math>
| value=<math>\tau\approx</math>0.41245403364...
| OEIS=A014571
| 發現者=
| define=<math>\tau = \sum_{i=0}^{\infty} \frac{t_i}{2^{i+1}}</math>， 其中<math>t_i</math>為{{link-en|蘇-摩爾斯數列|Thue–Morse sequence}}中的第''i''個元素。
| type=[[無理數]]<br/>[[超越數]]
| basedata = {{Infobox number/base
| 二進制 = {{FractionalGaps|{{進制|2|0.412454033640107597783361368258455|precision=24}}|4|…}}
| 十進位 = {{FractionalGaps|0.412454033640107597783361|4|…}}
| 十六進位 = {{FractionalGaps|{{進制|16|0.412454033640107597783361368258455|precision=24}}|4|…}} }}
| 連分數=[0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]
}}
'''普羅海特-蘇-摩爾斯常數'''（'''Prouhet–Thue–Morse constant'''）是[[數學]]中的[[常數]]，符號為<math>\tau</math>，得名自
{{link-fr|歐仁·普羅海特|Eugène Prouhet}}、[[阿克塞尔·图厄]]及{{link-en|馬斯頓·摩斯|Marston Morse}}，其[[二進制]].01101001100101101001011001101001...為{{link-en|蘇-摩爾斯數列|Thue–Morse sequence}}，也就是
: <math>  \tau = \sum_{i=0}^{\infty} \frac{t_i}{2^{i+1}} = 0.412454033640 \ldots </math>

其中<math>t_i</math>為蘇-摩爾斯數列中的第''i''個元素。

<math>t_i</math>的其生成級數為：
:<math> \tau(x) = \sum_{i=0}^{\infty} (-1)^{t_i} \, x^i  = \frac{1}{1-x} - 2 \sum_{i=0}^{\infty} t_i \, x^i</math>
可以表示為

: <math> \tau(x) = \prod_{n=0}^{\infty} ( 1 - x^{2^n} ). </math>
這是{{link-en|弗賓納斯多項式|Frobenius polynomial}}的乘積，因此可以推廣到任意的[[域 (數學)|域]]。

普羅海特-蘇-摩爾斯常數已由{{link-en|库尔特·马勒|Kurt Mahler}}在1929年證明是[[超越數]]<ref>{{cite journal | first=Kurt | last=Mahler | title=Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen | journal=Math. Annalen | volume=101 | year=1929 | pages=342–366 | jfm=55.0115.01 | doi=10.1007/bf01454845}}</ref>。

==腳註==
{{reflist}}

==參考資料==
*{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey  | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}.
* {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}

== 外部連結 ==
* {{SloanesRef |sequencenumber=A010060|name=Thue-Morse sequence}}
* [http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps The ubiquitous Prouhet-Thue-Morse sequence] {{Wayback|url=http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq.ps |date=20210225095311 }}, John-Paull Allouche and Jeffrey Shallit, (undated, 2004 or earlier) provides many applications and some history
* [http://planetmath.org/encyclopedia/ProuhetThueMorseConstant.html PlanetMath entry] {{Wayback|url=http://planetmath.org/encyclopedia/ProuhetThueMorseConstant.html |date=20120716181759 }}

{{無理數導航}}

{{DEFAULTSORT:Prouhet-Thue-Morse Constant}}
[[Category:數學常數]]
[[Category:数论]]
[[Category:实数超越数]]