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五邊形數定理

五邊形數定理是一個由歐拉發現的數學定理，描述歐拉函數 $\phi (q)$ 展開式的特性^[1] ^[2]。歐拉函數的展開式如下：

\prod _{n=1}^{\infty }(1-x^{n})=\sum _{k=-\infty }^{\infty }(-1)^{k}x^{\frac {k(3k-1)}{2}}=\sum _{k=0}^{\infty }(-1)^{k}x^{\frac {k(3k\pm 1)}{2}}

亦即

(1-x)(1-x^{2})(1-x^{3})\cdots =1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}+\cdots .

歐拉函數展開後，有些次方項被消去，只留下次方項為1, 2, 5, 7, 12, ...的項次，留下來的次方恰為廣義五邊形數。

若將上式視為幂級數，其收斂半徑為1，不過若只是當作形式冪級數來考慮，就不會考慮其收斂半徑。

和分割函數的關係

歐拉函數的倒數是分割函數的母函數，亦即：

{\frac {1}{\phi (x)}}=\sum _{k=0}^{\infty }p(k)x^{k}

其中 $p(k)$ 為k的分割函數。

上式配合五邊形數定理，可以得到

(1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}+\cdots )(1+p(1)x+p(2)x^{2}+p(3)x^{3}+\cdots )=1

考慮 $x^{n}$ 項的係數，在 n>0 時，等式右側的係數均為0，比較等式二側的係數，可得

p(n)-p(n-1)-p(n-2)+p(n-5)+p(n-7)+\cdots =0

因此可得到分割函數p(n)的递归式

p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots

以n=10為例

p(10)=p(9)+p(8)-p(5)-p(3)=30+22-7-3=42

參考資料

^ 原文為Euler, Leonhard. Evolutio producti infiniti $(1-x)(1-xx)(1-x^{3})(1-x^{4})(1-x^{5})$ etc. in seriem simplicem. Acta Academiae Scientarum Imperialis Petropolitinae. 1775, 1780: 47–55.
^ 英文翻譯版為Bell, J在2004-12-4翻譯的《The Expansion of the Infinite Product $(1-x)(1-xx)(1-x^{3})(1-x^{4})(1-x^{5})(1-x^{6})$ etc. into a Single Series》，http://www.arxiv.org/abs/math.HO/0411454/.

外部連結

Euler and the pentagonal number theorem
On Euler's Pentagonal Theorem（页面存档备份，存于互联网档案馆） at MathPages
Number Theorem.^{[永久失效連結]} at MathWorld
The Pentagonal Number Theorem and All That（页面存档备份，存于互联网档案馆） from Dick Koch.

查论编有形數
多邊形數	三角形數四邊形數五邊形數六邊形數七邊形數八邊形數九邊形數十邊形數十二邊形數
多面體數	四面體數六面體數八面體數
錐體數	三角錐數四角錐數五角錐數六角錐數七角錐數
中心多邊形數	中心三角形數中心四邊形數中心五邊形數中心六邊形數中心七邊形數中心十二邊形數
中心多面體數	中心四面體數中心六面體數中心八面體數
多胞體數	1-多胞體數 2-多胞體數 3-多胞體數 4-多胞體數
星數	五角星數六角星數七角星數八角星數六角星質數
其他有形數	平方数立方數四次方數五次方數六次方數三角平方數梯形數普洛尼克数
其他	费马多边形数定理四平方和定理五邊形數定理

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