Göllnitz-Gordon identities and Ramanujan-Göllnitz-Gordon continued fractions

수학노트
imported>Pythagoras0님의 2012년 10월 28일 (일) 16:45 판 (찾아 바꾸기 – “<h5 (.*)">” 문자열을 “==” 문자열로)
둘러보기로 가기 검색하러 가기

introduction

 

 

q-hypergeometric series

  • Slater 34
    \(\sum_{n=0}^{\infty}\frac{q^{n^2+2n}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=1/(q^{3};q^{8})_{\infty}(q^{4};q^{8})_{\infty}(q^{5};q^{8})_{\infty}}\)
  • Slater 36
    \(\sum_{n=0}^{\infty}\frac{q^{n^2}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=1/(q^{1};q^{8})_{\infty}(q^{4};q^{8})_{\infty}(q^{7};q^{8})_{\infty}}\)
  • lifting to rank 2 case

Using the Guass formula (in useful techniques in q-series)

\(\prod_{r=0}^{n-1}(1+zq^r)=(1+z)(1+zq)\cdots(1+zq^{n-1})= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r\)

we can rewrite \((-q;q^{2})_{n}\) as

\((-q;q^{2})_{n}=\sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r^{2}}\)

Therefore,

\(\sum_{n=0}^{\infty}\frac{q^{n^2}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^2}\sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r^{2}}}{ (q^{2};q^{2})_{n}}\)

\(=\sum_{i,j\geq 0}\frac{q^{(i+j)^2}q^{j^2}}{(-q)_{i+j}(q)_{j}(q)_{i}}=\sum_{i,j\geq 0}\frac{q^{i^2+2ij+2j^2}}{(-q)_{i+j}(q)_{j}(q)_{i}}\)

 

and

\(\sum_{n=0}^{\infty}\frac{q^{n^2+2n}(-q;q^{2})_{n}}{ (q^{2};q^{2})_{n}}=\sum_{i,j\geq 0}\frac{q^{i^2+2ij+2j^2}q^{2(i+j)}}{(-q)_{i+j}(q)_{j}(q)_{i}}\)

 

one can obtain the following matrix from the above hypergeometric series\[ \begin{bmatrix} 4 & 2 \\ 2 & 2 \end{bmatrix}\]

 

 

 

 

history

 

 

related items

 

 

encyclopedia

 

 

articles

원본 주소 "https://wiki.mathnt.net/index.php?title=Göllnitz-Gordon_identities_and_Ramanujan-Göllnitz-Gordon_continued_fractions&oldid=38915"