"슬레이터 목록 (Slater's list)"의 두 판 사이의 차이

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잔글 (찾아 바꾸기 – “==관련도서== * 도서내검색<br> ** http://books.google.com/books?q= ** http://book.daum.net/search/contentSearch.do?query=” 문자열을 “” 문자열로)
잔글 (찾아 바꾸기 – “<br><math>” 문자열을 “:<math>” 문자열로)
15번째 줄: 15번째 줄:
 
* '''[Slater51] '''(1.3)
 
* '''[Slater51] '''(1.3)
 
* '''[Slater51] '''(2.1)
 
* '''[Slater51] '''(2.1)
−
* '''[Slater51] '''(4.1)<br><math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
+
* '''[Slater51] '''(4.1):<math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
−
* '''[Slater51] '''(4.2)<br><math>\sum_{r=-[n/2]}^{r=[n/2]}\frac{(1-aq^{4r})(q^{-n})_{2r}a^{2r}q^{2nr+r}(d)_{q^2,r}(e)_{q^2,r}}{(1-a)(aq^{n+1})_{2r}d^re^r(aq^2/d)_{q^2,r}(aq^2/e)_{q^2,r}}=\frac{(q^2/a,aq/d,aq/e,aq^2/de;q^2)_{\infty}}{(q,q^2/d,q^2/e,a^2q/de;q^2)_{\infty}}\frac{(q)_{n}(aq)_{n}(a^2/de)_{q^2,n}}{(aq)_{q^2,n}(aq/d)_{n}(aq/e)_{n}}</math><br>
+
* '''[Slater51] '''(4.2):<math>\sum_{r=-[n/2]}^{r=[n/2]}\frac{(1-aq^{4r})(q^{-n})_{2r}a^{2r}q^{2nr+r}(d)_{q^2,r}(e)_{q^2,r}}{(1-a)(aq^{n+1})_{2r}d^re^r(aq^2/d)_{q^2,r}(aq^2/e)_{q^2,r}}=\frac{(q^2/a,aq/d,aq/e,aq^2/de;q^2)_{\infty}}{(q,q^2/d,q^2/e,a^2q/de;q^2)_{\infty}}\frac{(q)_{n}(aq)_{n}(a^2/de)_{q^2,n}}{(aq)_{q^2,n}(aq/d)_{n}(aq/e)_{n}}</math><br>
 
* '''[Slater51] '''(4.3)
 
* '''[Slater51] '''(4.3)
  
25번째 줄: 25번째 줄:
 
==Group B==
 
==Group B==
  
−
* '''[Slater51] '''(4.1)<br><math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
+
* '''[Slater51] '''(4.1):<math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
 
*  B(1)<br>
 
*  B(1)<br>
 
** [[슬레이터 18]]
 
** [[슬레이터 18]]
48번째 줄: 48번째 줄:
 
==Group H==
 
==Group H==
  
−
*   '''[Slater51] '''(4.1)<br><math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
+
*   '''[Slater51] '''(4.1):<math>\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}</math><br>
  
 
* [[슬레이터 1]]
 
* [[슬레이터 1]]
58번째 줄: 58번째 줄:
 
==슬레이터 목록==
 
==슬레이터 목록==
  
−
* [[슬레이터 1]]<br><math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n=1}^{\infty}(-1)^{n}(q^{\frac{3 n^2-n}{2}}+q^{\frac{3 n^2+n}{2}})=\sum_{n=-\infty}^\infty(-1)^nq^{n(3n-1)/2}</math><br>
+
* [[슬레이터 1]]:<math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n=1}^{\infty}(-1)^{n}(q^{\frac{3 n^2-n}{2}}+q^{\frac{3 n^2+n}{2}})=\sum_{n=-\infty}^\infty(-1)^nq^{n(3n-1)/2}</math><br>
−
* [[슬레이터 2]]<br><math>\prod_{n=1}^{\infty}(1+q^n)=\sum_{n=1}^{\infty}\frac{q^{n(n+1)/2}}{(q)_n}</math><br>
+
* [[슬레이터 2]]:<math>\prod_{n=1}^{\infty}(1+q^n)=\sum_{n=1}^{\infty}\frac{q^{n(n+1)/2}}{(q)_n}</math><br>
−
* [[슬레이터 8]]<br><math>\sum_{n=0}^{\infty}\frac{(q^2;q^2)_{n}q^{n(n+1)/2}}{ (q)_{n}^2}=\frac{(-q)_{\infty}}{(q^2;q^4)_{\infty}}</math><br>
+
* [[슬레이터 8]]:<math>\sum_{n=0}^{\infty}\frac{(q^2;q^2)_{n}q^{n(n+1)/2}}{ (q)_{n}^2}=\frac{(-q)_{\infty}}{(q^2;q^4)_{\infty}}</math><br>
  
 
 
 
 

2013년 1월 12일 (토) 09:56 판

이 항목의 수학노트 원문주소

 

 

개요

 

 

주요 항등식

  • [Slater51] (1.3)
  • [Slater51] (2.1)
  • [Slater51] (4.1)\[\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}\]
  • [Slater51] (4.2)\[\sum_{r=-[n/2]}^{r=[n/2]}\frac{(1-aq^{4r})(q^{-n})_{2r}a^{2r}q^{2nr+r}(d)_{q^2,r}(e)_{q^2,r}}{(1-a)(aq^{n+1})_{2r}d^re^r(aq^2/d)_{q^2,r}(aq^2/e)_{q^2,r}}=\frac{(q^2/a,aq/d,aq/e,aq^2/de;q^2)_{\infty}}{(q,q^2/d,q^2/e,a^2q/de;q^2)_{\infty}}\frac{(q)_{n}(aq)_{n}(a^2/de)_{q^2,n}}{(aq)_{q^2,n}(aq/d)_{n}(aq/e)_{n}}\]
  • [Slater51] (4.3)

 

 

Group B

  • [Slater51] (4.1)\[\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}\]
  • B(1)
  • B(2)

 

 

 

Group E

 

 

Group H

  •  [Slater51] (4.1)\[\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}\]

 

 

슬레이터 목록

  • 슬레이터 1\[\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n=1}^{\infty}(-1)^{n}(q^{\frac{3 n^2-n}{2}}+q^{\frac{3 n^2+n}{2}})=\sum_{n=-\infty}^\infty(-1)^nq^{n(3n-1)/2}\]
  • 슬레이터 2\[\prod_{n=1}^{\infty}(1+q^n)=\sum_{n=1}^{\infty}\frac{q^{n(n+1)/2}}{(q)_n}\]
  • 슬레이터 8\[\sum_{n=0}^{\infty}\frac{(q^2;q^2)_{n}q^{n(n+1)/2}}{ (q)_{n}^2}=\frac{(-q)_{\infty}}{(q^2;q^4)_{\infty}}\]

 

 

역사

 

 

 

메모

 

 

 

관련된 항목들

 

 

수학용어번역

 

 

 

사전 형태의 자료

 

 

리뷰논문, 에세이, 강의노트

 

 

 

관련논문

 

 

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