"합공식의 q-analogue"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
3번째 줄: 3번째 줄:
 
* [[초기하급수의 합공식|초기하 급수의 합공식]]
 
* [[초기하급수의 합공식|초기하 급수의 합공식]]
 
* [[q-Chu-Vandermonde 항등식|q-Chu-Vandermonde]]:<math>_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n</math>
 
* [[q-Chu-Vandermonde 항등식|q-Chu-Vandermonde]]:<math>_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n</math>
−
* '''[GR2004]''' (1.5.1) Heine's q-analogue of Gauss' summation formula:<math>_2\phi_1(a,b;c,q,c/ab)=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}</math> or :<math>\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}</math>
+
* '''[GR2004]''' (1.5.1) Heine's q-analogue of Gauss' summation formula:<math>_2\phi_1(a,b;c,q,c/ab)=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}</math> or :<math>\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}</math>
−
* '''[GR2004]''' (1.7.2) q-analogue of Pfaff-Saalschutz's summation formula:<math>_3\phi_2(a,b,q^{-n};c,abc^{-1}q^{1-n};q,q)=\frac{(c/a,c/b;q)_{n}}{(c,c/ab;q)_{n}}</math> or:<math>\sum_{n=0}^{\infty}\frac{q^n (a;q)_n (b;q)_n \left(q^{-k};q\right)_n}{(q;q)_n (c;q)_n \left(\frac{a b q^{1-k}}{c};q\right)_n}=\frac{\left(\frac{c}{a};q\right)_k \left(\frac{c}{b};q\right)_k}{(c;q)_k \left(\frac{c}{a b};q\right)_k}</math>
+
* '''[GR2004]''' (1.7.2) q-analogue of Pfaff-Saalschutz's summation formula:<math>_3\phi_2(a,b,q^{-n};c,abc^{-1}q^{1-n};q,q)=\frac{(c/a,c/b;q)_{n}}{(c,c/ab;q)_{n}}</math> or:<math>\sum_{n=0}^{\infty}\frac{q^n (a;q)_n (b;q)_n \left(q^{-k};q\right)_n}{(q;q)_n (c;q)_n \left(\frac{a b q^{1-k}}{c};q\right)_n}=\frac{\left(\frac{c}{a};q\right)_k \left(\frac{c}{b};q\right)_k}{(c;q)_k \left(\frac{c}{a b};q\right)_k}</math>
 
*  q-analogue of Whipple's theorem
 
*  q-analogue of Whipple's theorem
 
* [[Jackson's q-analogue of Dougall's theorem]]
 
* [[Jackson's q-analogue of Dougall's theorem]]
  
−
 
+
  
−
 
+
  
−
 
+
  
−
 
+
  
 
==== 하위페이지 ====
 
==== 하위페이지 ====
23번째 줄: 23번째 줄:
 
** [[q-가우스 합]]
 
** [[q-가우스 합]]
  
−
 
+
  
−
 
+
  
−
 
+
  
 
==books==
 
==books==

2020년 12월 28일 (월) 04:11 기준 최신판

introduction

  • 초기하 급수의 합공식
  • q-Chu-Vandermonde\[_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n\]
  • [GR2004] (1.5.1) Heine's q-analogue of Gauss' summation formula\[_2\phi_1(a,b;c,q,c/ab)=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\] or \[\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\]
  • [GR2004] (1.7.2) q-analogue of Pfaff-Saalschutz's summation formula\[_3\phi_2(a,b,q^{-n};c,abc^{-1}q^{1-n};q,q)=\frac{(c/a,c/b;q)_{n}}{(c,c/ab;q)_{n}}\] or\[\sum_{n=0}^{\infty}\frac{q^n (a;q)_n (b;q)_n \left(q^{-k};q\right)_n}{(q;q)_n (c;q)_n \left(\frac{a b q^{1-k}}{c};q\right)_n}=\frac{\left(\frac{c}{a};q\right)_k \left(\frac{c}{b};q\right)_k}{(c;q)_k \left(\frac{c}{a b};q\right)_k}\]
  • q-analogue of Whipple's theorem
  • Jackson's q-analogue of Dougall's theorem





하위페이지




books

원본 주소 "https://wiki.mathnt.net/index.php?title=합공식의_q-analogue&oldid=48327"