Bailey pair and lemma

 

 

 

 

 

 

 

Bailey pair

• the sequence \(\{\alpha_r\}, \{\beta_r\}\) satisfying the following is called a Bailey pair
  \(\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}\)
  
• conjugate Bailey pair  \(\{\delta_r\}, \{\gamma_r\}\)
  \(\gamma_L=\sum_{r=L}^{\infty}\frac{\delta_r}{(q)_{r-L}(aq)_{r+L}}\)
  

 

 

Bailey chain

• we derive a new Bailey chain from a known Bailey pair
  \(\alpha^\prime_n= \frac{(\rho_1;q)_n(\rho_2;q)_n(aq/\rho_1\rho_2)^n}{(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\alpha_n\)
  \(\beta^\prime_n = \sum_{j\ge0}\frac{(\rho_1;q)_j(\rho_2;q)_j(aq/\rho_1\rho_2;q)_{n-j}(aq/\rho_1\rho_2)^j}{(q;q)_{n-j}(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\beta_j\)
  
• corollary. by taking  \(\rho_1,\rho_2\to \infty\) , we get 
   
  \(\alpha^\prime_n= a^nq^{n^2}\alpha_n\)
  \(\beta^\prime_n = \sum_{r=0}^{L}\frac{a^rq^{r^2}}{(q)_{L-r}}\beta_j\)
  

 

 

history

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

 

 

 

related items

• Bloch group
  
• Bloch group, K-theory and dilogarithm
  
• manufacturing matrices from lower ranks
  

 

 

encyclopedia

•  
  
• http://en.wikipedia.org/wiki/Bailey_pair
• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

[[4909919|]]

 

 

articles

•  
  
• 50 Years of Bailey's lemma
  
  • S. Ole Warnaar, 2009
    
• Andrews–Gordon type identities from combinations of Virasoro characters
  
  • Boris Feigin, Omar Foda, Trevor Welsh, 2007
    
•  
  Finite Rogers-Ramanujan Type Identities
  
  • Andrew V. Sills, 2003
    
• Rogers-Ramanujan-Slater Type identities
  
  • Mc Laughlin
    
• Virasoro character identities from the Andrews–Bailey construction
  
  • Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)
    
• Further identities of the Rogers-Ramanujan type
  
  • Slater, L. J. (1952),  Proceedings of the London Mathematical Society. Second Series 54: 147–167
    

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html[1]
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

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

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/