"Durfee 사각형 항등식(Durfee rectangle identity)"의 두 판 사이의 차이
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| − | * (Durfee rectangle identity)<br> | + | * (Durfee rectangle identity)<br><math>l \in \mathbb{Z}</math>,<br><math>\sum_{n,m\geq 0, n-m=l}\frac{q^{nm}}{(q)_n(q)_m}=\frac{1}{(q)_{\infty}}</math> 또는<br><math>\sum_{n\geq 0}\frac{q^{n(n+l)}}{(q)_n(q)_{n+l}}=\frac{1}{(q)_{\infty}}</math><br> |
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| + | <math>\frac{\theta}{\eta}=\frac{\sum_{l\geq 0}q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{n,m\geq 0}\frac{q^{\frac{1}{2}(an^2+(2-2a)mn+am^2)+b(n-m)+c}}{(q)_n(q)_m}</math> | ||
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| + | (pf) | ||
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| + | <math>\frac{\sum_{l\geq 0}q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{l\geq 0}\frac{q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{l\geq 0}\sum_{n,m\geq 0, n-m=l}\frac{q^{\frac{a}{2}l^2+bl+c}q^{nm}}{(q)_n(q)_m}\sum_{n,m\geq 0}\frac{q^{nm+\frac{a}{2}(n-m)^2+b(n-m)+c}}{(q)_n(q)_m}=\sum_{n,m\geq 0}\frac{q^{\frac{1}{2}(an^2+(2-2a)mn+am^2)+b(n-m)+c}}{(q)_n(q)_m}</math> | ||
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| + | where we used <math>l=n-m</math>. ■ | ||
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| + | (따름정리) | ||
<math>\sum_{n=0}^\infty p(n)q^n = 1+\sum_{n=1}\frac{q^{n^2}}{(1-q)^2(1-q^2)^2\cdots(1-q^n)^2}</math> | <math>\sum_{n=0}^\infty p(n)q^n = 1+\sum_{n=1}\frac{q^{n^2}}{(1-q)^2(1-q^2)^2\cdots(1-q^n)^2}</math> | ||
2011년 7월 29일 (금) 07:53 판
- (Durfee rectangle identity)
\(l \in \mathbb{Z}\),
\(\sum_{n,m\geq 0, n-m=l}\frac{q^{nm}}{(q)_n(q)_m}=\frac{1}{(q)_{\infty}}\) 또는
\(\sum_{n\geq 0}\frac{q^{n(n+l)}}{(q)_n(q)_{n+l}}=\frac{1}{(q)_{\infty}}\)
\(\frac{\theta}{\eta}=\frac{\sum_{l\geq 0}q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{n,m\geq 0}\frac{q^{\frac{1}{2}(an^2+(2-2a)mn+am^2)+b(n-m)+c}}{(q)_n(q)_m}\)
(pf)
\(\frac{\sum_{l\geq 0}q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{l\geq 0}\frac{q^{\frac{a}{2}l^2+bl+c}}{(q)_{\infty}}=\sum_{l\geq 0}\sum_{n,m\geq 0, n-m=l}\frac{q^{\frac{a}{2}l^2+bl+c}q^{nm}}{(q)_n(q)_m}\sum_{n,m\geq 0}\frac{q^{nm+\frac{a}{2}(n-m)^2+b(n-m)+c}}{(q)_n(q)_m}=\sum_{n,m\geq 0}\frac{q^{\frac{1}{2}(an^2+(2-2a)mn+am^2)+b(n-m)+c}}{(q)_n(q)_m}\)
where we used \(l=n-m\). ■
(따름정리)
\(\sum_{n=0}^\infty p(n)q^n = 1+\sum_{n=1}\frac{q^{n^2}}{(1-q)^2(1-q^2)^2\cdots(1-q^n)^2}\)
(증명)
http://cfranc.wordpress.com/2009/11/24/an-identity-of-ramanujan/ ■
http://www.springerlink.com/content/l842207736576587/
http://siba-ese.unisalento.it/index.php/quadmat/article/download/6953/6317