Bailey pair and lemma

introduction

•  q-Pfaff-Sallschutz sum
  

 

 

Bailey lemma

If the sequence \(\{\alpha_r\}, \{\beta_r\}\), \(\{\delta_r\}, \{\gamma_r\}\) satisfy the following

\(\beta_L=\sum_{r=0}^{L}{\alpha_r}{u_{L-r}v_{L+r}}\), \(\gamma_L=\sum_{r=L}^{\infty}{\delta_r}{u_{r-L}v_{r+L}}\)

then,

\(\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}\)

 

(corolary 1)

Choose the following

\(u_{n}=\frac{1}{(q)_n}\) ,\(v_{n}=\frac{1}{(x)_n}\),\(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\)

Then 

 \(\gamma_n=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\),

and hence by Bailey's lemma,

\(\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}\)

(proof)

By the basic analogue of Gauss' theorem 

(Recall \(\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\), q-analogue of summation formulas )

Also note that \((a)_{n+r}=(a)_{n}(aq^{n})_{r}\). 

Put \(a=yq^{n},b=zq^{n},c=xq^{2n}\). 

\(\gamma_n=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\) (a different notation \(\gamma_n=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\) is also used sometimes) ■

 

 

 

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

• Bailey chain
  

 

 

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/Wilfrid_Norman_Bailey
• 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
    
• A generalization of the q-Saalschutz sum and the Burge transform
  
  • A. Schilling, S.O. Warnaa, 2009
    
• Rogers-Ramanujan-Slater Type identities
  
  • Mc Laughlin, 2008
    
• 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
    
• Virasoro character identities from the Andrews–Bailey construction
  
  • Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)
    

• Multiple series Rogers-Ramanujan type identities.
  
  • George E. Andrews, Pacific J. Math.  Volume 114, Number 2 (1984), 267-283.
    
• Special values of the dilogarithm function
  
  • J. H. Loxton, 1984
• Wilfrid Norman Bailey
  
  • Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512
    
• Further identities of the Rogers-Ramanujan type
  
  • Slater, L. J. (1952),  Proceedings of the London Mathematical Society. Second Series 54: 147–167
    
• A New Proof of Rogers's Transformations of Infinite Series
  
  • Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475
    
• Identities of Rogers-Ramanujan type
  
  • Bailey, 1944
    
• On two theorems of combinatory analysis and some allied identities 
  
• 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/10.1112/plms/s2-53.6.460

 

 

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/