베일리 사슬(Bailey chain)

이 항목의 수학노트 원문주소

• 베일리 사슬(Bailey chain)

 

 

 

 

 

베일리 사슬

•  
  기존의 베일리 쌍 relative to a  로부터 새로운 베일리 쌍 relative to a 을 얻는 방법
  \(\alpha^\prime_n= \frac{(\rho_1;q)_n(\rho_2;q)_n(aq/\rho_1\rho_2)^n}{(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\alpha_n\)
  \(\beta^\prime_n = \sum_{j=0}^{n}\frac{(\rho_1;q)_j(\rho_2;q)_j(aq/\rho_1\rho_2;q)_{n-j}(aq/\rho_1\rho_2)^j}{(q;q)_{n-j}(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\beta_j\)
  

• 위에서  \(\rho_1,\rho_2\to \infty\) 일 경우, 다음을 얻는다
  \(\alpha^\prime_n= a^nq^{n^2}\alpha_n\)
  \(\beta^\prime_n = \sum_{r=0}^{n}\frac{a^rq^{r^2}}{(q)_{n-r}}\beta_r\)
  
• applying the Bailey pair equation
  \(\beta^{'}_n=\sum_{r=0}^{n}\frac{\alpha^{'}_r}{(q)_{n-r}(aq)_{n+r}}\)
  to the new Bailey pair, we get
  \(\sum_{r=0}^{n}\frac{a^{r}q^{r^2}\alpha_r}{(q)_{n-r}(aq)_{n+r}}= \sum_{r=0}^{n}\frac{a^rq^{r^2}}{(q)_{n-r}}\beta_r\)
  

 

 

repetition of chain construction

• repeating this construction several times
  \(\sum_{r=0}^{n}\frac{a^{kr}q^{kr^2}\alpha_r}{(q)_{n-r}(aq)_{n+r}}=\sum_{n_1=0}^{n}\sum_{n_2=0}^{n_1}\cdots\sum_{n_k=0}^{n_{k-1}}\frac{a^{n_1+\cdots+n_{k}}q^{n_1^2+\cdots+n_{k}^2}\beta_{n_{k}}}{(q)_{n-n_{1}}(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}-n_{k}}}\)
  

• Taking \(n\to\infty\), we get
  \(\frac{1}{(aq)_{\infty}}\sum_{n=0}^{\infty}a^{kn}q^{kn^{2}}\alpha_{n}=\sum_{n_1\geq\cdots\geq n_{k}\geq0}\frac{a^{n_1+\cdots+n_{k}}q^{n_1^2+\cdots+n_{k}^2}\beta_{n_{k}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}-n_{k}}}\)
  

 

 

examples

• initial Bailey pair
  \(\alpha_{L}=(-1)^{L}q^{\binom{L}{2}}\frac{(1-aq^{2L})(a)_{L}}{(1-a)(q)_{L}}=(-1)^{L}q^{L(L-1)/2}\frac{(1-aq^{2L})(a)_{L}}{(1-a)(q)_{L}}\)
  \(\beta_{L}=\delta_{L,0}\)
  For example, if a=1,
  \(\alpha_{L}=(-1)^{L}q^{L(L-1)/2}(1+q^{L})=(-1)^{L}(q^{(3L^2-L)/2}+q^{(3L^2+L)/2})\)
  
• result of Bailey chain applied k-times
  \(\alpha_{L}=(-1)^{L}a^{kL}q^{kL^{2}+L^2/2-L/2}\frac{(1-aq^{2L})(a)_{L}}{(1-a)(q)_{L}}\)
  \(\beta_{L}=\sum_{L\geq n_1\geq\cdots\geq n_{k-1}\geq0}\frac{a^{n_1+\cdots+n_{k-1}}q^{n_1^2+\cdots+n_{k-1}^2}}{(q)_{L-n_1}(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}\)
  
• obtained q-series identity
  \(\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r((2k+1)r+1-2jk)/2}=\sum_{n_1\geq\cdots\geq n_{k}\geq0}\frac{q^{n_1^2+\cdots+n_{k}^2+j(n_1+\cdots+n_{k})}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}\)
  
• Setting k=1, a=1, we get the Euler pentagonal number theorem
  \((q)_{\infty}=\sum_{k=-\infty}^\infty(-1)^kq^{k(3k-1)/2}\)
  
• Setting k=2, a=1, we get one of RR identity
  \(\sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} \)
  
• Setting k=2, a=q, we get one of RR identity
   \(\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}\)
  
• We frequently use Jacobi triple product identity
  \(\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\)
  

• if k is bigger than 2, we get some cases of 앤드류스-고든 항등식(Andrews-Gordon identity)
  
• 모든 앤드류스-고든 항등식(Andrews-Gordon identity)  을 증명하려면, 베일리 격자(Bailey lattice) 가 필요하다
  

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표

 

 

메모

 

• Math Overflow http://mathoverflow.net/search?q=

 

 

 

 

수학용어번역

• 단어사전
  
  • http://translate.google.com/#en%7Cko%7C
  • http://ko.wiktionary.org/wiki/
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 한국통계학회 통계학 용어 온라인 대조표
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

 

 

 

 

 

관련논문

• Paule, The Concept of Bailey Chains
• George E. AndrewsMultiple series Rogers-Ramanujan type identities., Pacific J. Math.  Volume 114, Number 2 (1984), 267-283.
  

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=