Lebesgue identity - 편집 역사

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2020-11-16T14:09:39Z

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2020-11-13T13:44:53Z

imported>Pythagoras0: /* related items */

2020-11-13T13:44:43Z

imported>Pythagoras0: /* articles */

2020-11-13T13:44:32Z

articles

2013년 7월 15일 (월) 11:54에 imported>Pythagoras0님의 편집

2013-07-15T11:54:26Z

imported>Pythagoras0: /* fermionic formula */

2013-05-13T09:56:54Z

fermionic formula

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2013-04-24T14:07:09Z

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imported>Pythagoras0: Pythagoras0 사용자가 Lebesgue's identity 문서를 Lebesgue identity 문서로 옮겼습니다.

2013-04-21T13:09:21Z

Pythagoras0 사용자가 Lebesgue's identity 문서를 Lebesgue identity 문서로 옮겼습니다.

@@ 151번째 줄: / 151번째 줄: @@
 [[분류:개인노트]]
 [[분류:q-series]]
 [[분류:migrate]]

@@ 43번째 줄: / 43번째 줄: @@
 ==specializations==
 * we expect to find five vectors for linear terms
-$$\vec{b}=(1/2,-1),(0,0),(1/2,0),(1/2,1),(1,1)$$
+:<math>\vec{b}=(1/2,-1),(0,0),(1/2,0),(1/2,1),(1,1)</math>
 * for a complete list, see [[Ramanujan-Göllnitz-Gordon continued fraction]]
 ===Theorem===
-For $\vec{b}=(1/2,0)$,
+For <math>\vec{b}=(1/2,0)</math>,
 :<math>f(q,q^{-1})=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}}}{(q)_{i}(q)_{j}}=(-q;q^2)_{\infty}(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}^3}{(q;q)_{\infty}^2(q^{4};q^{4})_{\infty}}=\frac{(q^2;q^4)_{\infty}}{(q;q^4)_{\infty}^2(q^3;q^4)_{\infty}^2},</math>
-For $\vec{b}=(1/2,1)$,
+For <math>\vec{b}=(1/2,1)</math>,
 :<math>f(q,1)=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}+j}}{(q)_{i}(q)_{j}}=(-q^2;q^2)_{\infty}(-q)_{\infty}=\frac{(q^4;q^4)_{\infty}}{(q;q)_{\infty}}=\frac{1}{(q^1;q^4)_{\infty}(q^2;q^4)_{\infty}(q^3;q^4)_{\infty}}</math>

@@ 133번째 줄: / 133번째 줄: @@
 ==related items==
 * [[3-states Potts model]]
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxZjNHc0hNUmxfREk/edit
@@ 148번째 줄: / 151번째 줄: @@
 [[분류:개인노트]]
 [[분류:q-series]]

@@ 133번째 줄: / 133번째 줄: @@
 ==related items==
 * [[3-states Potts model]]
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxZjNHc0hNUmxfREk/edit

← 이전 판		2020년 11월 13일 (금) 13:44 판
151번째 줄:		151번째 줄:

	[[분류:개인노트]]		[[분류:개인노트]]
−	~~[[분류:thesis]]~~
	[[분류:q-series]]		[[분류:q-series]]

@@ 20번째 줄: / 20번째 줄: @@
 \end{aligned}
 </math>
-By putting <math>j=r</math> and <math>k=i+j</math>,
+By putting <math>r=j</math> and <math>k=i+j</math>,
 :<math>
 \begin{aligned}
@@ 30번째 줄: / 30번째 줄: @@
 *  here we get a 2x2 matrix ([[rank 2 case]])
 :<math> \begin{bmatrix} 1 & 1  \\ 1 & 2 \end{bmatrix}</math>
 ==specilizations : Lebesgue's identity==

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * {{수학노트|url=르벡_항등식}}
-==fermionic form expression==
+==fermionic formula==
 :<math>f(a,z)=\sum_{k\geq 0}\frac{a^{k}q^{k(k-1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{a^{i+j}z^{j}q^{\frac{i^2+2ij+2j^2-i}{2}}}{(q)_{i}(q)_{j}}\label{faz}</math>
@@ 12번째 줄: / 10번째 줄: @@
 We use the q-binomial identity (see [[useful techniques in q-series]])
-:<math>(-z;q)_{n}= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r</math> and
+:<math>(-z;q)_{n}= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r</math> and
 :<math>(-zq;q)_{k}= \sum_{r=0}^{k} \begin{bmatrix} k\\ r\end{bmatrix}_{q}q^{r(r+1)/2}z^r.</math>
@@ 28번째 줄: / 26번째 줄: @@
 {}&=\sum_{i,j\geq 0}\frac{a^{i+j}z^{j}q^{\frac{i^2+2ij+2j^2-i}{2}}}{(q)_{i}(q)_{j}}
 \end{aligned}
-</math>  ■
+</math>  ■
-*  here we get a 2x2 matrix ([[rank 2 case]])
+*  here we get a 2x2 matrix ([[rank 2 case]])
 :<math> \begin{bmatrix} 1 & 1  \\ 1 & 2 \end{bmatrix}</math>
-−
-−
 *  Put a=q, c=z. we get Lebesgue's identity.
-:<math>f(q,z)=\sum_{k\geq 0}\frac{q^{k}q^{k(k-1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{k\geq 0}\frac{q^{k(k+1)/2}(-zq)_{k}}{(q)_{k}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}=\prod_{m=1}^{\infty} (1+zq^{2m})(1+q^{m})</math><br>
+:<math>f(q,z)=\sum_{k\geq 0}\frac{q^{k}q^{k(k-1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{k\geq 0}\frac{q^{k(k+1)/2}(-zq)_{k}}{(q)_{k}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}=\prod_{m=1}^{\infty} (1+zq^{2m})(1+q^{m})</math>
 *  special case : we get a rank 2 form of Lebesgue's identity
-:<math>f(q,z)=\sum_{k\geq 0}\frac{q^{k}q^{k(k-1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{z^{j}q^{\frac{i^2+2ij+j^2+i+2j}{2}}}{(q)_{i}(q)_{j}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}</math><br>
+:<math>f(q,z)=\sum_{k\geq 0}\frac{q^{k}q^{k(k-1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{z^{j}q^{\frac{i^2+2ij+j^2+i+2j}{2}}}{(q)_{i}(q)_{j}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}</math>
-−
-−
 ==specializations==
@@ 89번째 줄: / 86번째 줄: @@
 $$
 where $Q^2=q$
-* $\frac{\eta(4\tau)}{\eta(\tau)}$
+* the right hand sides of the last two identities also show up in [[Ramanujan-Göllnitz-Gordon continued fraction]]
 ==continued fraction expression==
-* [[rank 2 continued fraction]]<br>
+* [[rank 2 continued fraction]]
 * '''[Alladi&Gordon1993] 277-278p'''
 * Let <math>f(a,c)=\sum_{k\geq 0}\frac{a^{k}q^{k(k-1)/2}(-cq)_{k}}{(q)_{k}}</math> as above
 * consider the following continued fractions
 :<math>F(a,c)=\frac{f(a,c)}{f(aq,c)}=1+a+\frac{acq}{1+aq} {\ \atop+} \frac{acq^2}{1+aq^2}{\ \atop+} \frac{acq^3}{1} {\ \atop+\dots}</math>
-:<math>R(a,b)=\frac{f(a,a^{-1}b)}{f(aq,a^{-1}b)}-a=\frac{R^{N}(a,b)}{R^{D}(a,b)}=1+\frac{bq}{1+aq} {\ \atop+} \frac{bq^2}{1+aq^2}{\ \atop+} \frac{bq^3}{1} {\ \atop+\dots}</math><br> where
+:<math>R(a,b)=\frac{f(a,a^{-1}b)}{f(aq,a^{-1}b)}-a=\frac{R^{N}(a,b)}{R^{D}(a,b)}=1+\frac{bq}{1+aq} {\ \atop+} \frac{bq^2}{1+aq^2}{\ \atop+} \frac{bq^3}{1} {\ \atop+\dots}</math> where
 :<math>R^{N}(a,b)=f(q,a^{-1}b)-af(aq,a^{-1}b)=f(aq,a^{-1}bq^{-1})=\sum_{k\geq 0}\frac{a^{k}q^{k(k+1)/2}(-a^{-1}b)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}}}{(q)_{i}(q)_{j}}</math>
 and
-:<math>R^{D}(a,b)=f(aq,a^{-1}b)=\sum_{k\geq 0}\frac{a^{k}q^{k(k+1)/2}(-a^{-1}bq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}+j}}{(q)_{i}(q)_{j}}</math><br>
+:<math>R^{D}(a,b)=f(aq,a^{-1}b)=\sum_{k\geq 0}\frac{a^{k}q^{k(k+1)/2}(-a^{-1}bq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}+j}}{(q)_{i}(q)_{j}}</math>
 *  applications
 :<math>R^N(1,1)=f(q,q^{-1})=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}}}{(q)_{i}(q)_{j}}=(-q;q^2)_{\infty}(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}^3}{(q;q)_{\infty}^2(q^{4};q^{4})_{\infty}}=\frac{(q^2;q^4)_{\infty}}{(q^1;q^4)_{\infty}^2(q^3;q^4)_{\infty}^2}</math>
-:<math>R^{D}(1,1)=f(q,1)=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}+j}}{(q)_{i}(q)_{j}}=(-q^2;q^2)_{\infty}(-q)_{\infty}=\frac{(q^4;q^4)_{\infty}}{(q;q)_{\infty}}=\frac{1}{(q^1;q^4)_{\infty}(q^2;q^4)_{\infty}(q^3;q^4)_{\infty}}</math><br>
+:<math>R^{D}(1,1)=f(q,1)=\sum_{i,j\geq 0}\frac{q^{\frac{i^2+2ij+2j^2}{2}+\frac{i}{2}+j}}{(q)_{i}(q)_{j}}=(-q^2;q^2)_{\infty}(-q)_{\infty}=\frac{(q^4;q^4)_{\infty}}{(q;q)_{\infty}}=\frac{1}{(q^1;q^4)_{\infty}(q^2;q^4)_{\infty}(q^3;q^4)_{\infty}}</math>
 *  continued fraction
 :<math>R(1,1)=\frac{R^{N}(1,1)}{R^{D}(1,1)}=1+{q \over 1+q + } {q^2 \over 1+q^2+} {q^3 \over 1+q^3}  \cdots=\frac{(q^2;q^4)_{\infty}^2}{(q^1;q^4)_{\infty}(q^3;q^4)_{\infty}}</math>
-==related dilogarithm identity==
 ==comparison with Rogers-Selberg identities==
@@ 124번째 줄: / 115번째 줄: @@
 * [[Rogers-Selberg identities]]
 :<math>AG_{3,3}(q)=\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}</math>
-:<math>A(q)W(q)=AG_{3,3}(q)</math><br> where
+:<math>A(q)W(q)=AG_{3,3}(q)</math> where
-:<math>W(q)=(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}</math><br>
+:<math>W(q)=(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}</math>
 *  Lebesgue's identity
-:<math>\frac{W(q)^2}{W(q^2)}=\sum_{i,j\geq 0}\frac{q^{(i^2+2ij+2j^2)/2+i/2}}{(q)_{i}(q)_{j}}</math><br>
+:<math>\frac{W(q)^2}{W(q^2)}=\sum_{i,j\geq 0}\frac{q^{(i^2+2ij+2j^2)/2+i/2}}{(q)_{i}(q)_{j}}</math>
-−
 (proof)
@@ 139번째 줄: / 130번째 줄: @@
 :<math>(-q;q^2)_{\infty}(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}^3}{(q;q)_{\infty}^2(q^{4};q^{4})_{\infty}}=\frac{W(q)^2}{W(q^2)}</math>. ■
-−
-−
 :<math>W(q)=\frac{\eta(2\tau)}{\eta(\tau)}</math>
@@ 150번째 줄: / 141번째 줄: @@
 * see [[eta product and eta quotient]] also
-−
-−
 ==history==
@@ 158번째 줄: / 149번째 줄: @@
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
-−
-−
 ==related items==
-* [[1 Nahm's conjecture|Nahm's conjecture]]<br>
+* [[1 Nahm's conjecture|Nahm's conjecture]]
-* [[3-states Potts model]]<br>
+* [[3-states Potts model]]
-−
 ==computational resource==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxZjNHc0hNUmxfREk/edit
-* http://en.wikipedia.org/wiki/
 ==articles==

← 이전 판	2013년 4월 21일 (일) 23:45 판
168번째 줄:	168번째 줄:


−	+	==computational resource==
	+	* https://docs.google.com/file/d/0B8XXo8Tve1cxZjNHc0hNUmxfREk/edit

← 이전 판	2013년 4월 21일 (일) 13:09 판
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