Lebesgue identity

introduction

• [Alladi&Gordon1993] 278&279p
  
• Lebesgue's identity
  \(\sum_{k\geq 0}\frac{q^{k(k+1)/2}(-zq)_{k}}{(q)_{k}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}=\prod_{m=1}^{\infty} (1+zq^{2m})(1+q^{m})\)
  

 

 

a 2x2 matrix

• Use q-binomial identity
   \((-z;q)_{n}= \sum_{r=0}^{n} \begin{bmatrix} n\\ r\end{bmatrix}_{q}q^{r(r-1)/2}z^r\) and \((-zq;q)_{k}= \sum_{r=0}^{k} \begin{bmatrix} k\\ r\end{bmatrix}_{q}q^{r(r+1)/2}z^r\)
  
• we get a rank 2 form of the Lebesgue's identity
  \(\sum_{k\geq 0}\frac{q^{k(k+1)/2}(-zq)_{k}}{(q)_{k}}=\sum_{i,j\geq 0}\frac{z^{j}q^{(i+j)(i+j+1)/2+j(j+1)/2}}{(q)_{i}(q)_{j}}=(-zq^2;q^2)_{\infty}(-q)_{\infty}\) where \(i=k-j\).
  

• here we get a 2x2 matrix (rank 2 case)
  \( \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}\)
  

 

 

specializations

•  
  From the above, we can derive
  \(\sum_{i,j\geq 0}\frac{q^{(i^2+2ij+2j^2)/2+i/2}}{(q)_{i}(q)_{j}}=(-q;q^2)_{\infty}(-q)_{\infty}\)
  

• Slater's list
  

 

 

comparison with Rogers-Selberg identities

• Rogers-Selberg identities
  \(AG_{3,3}(q)=\sum_{n_1,n_{2}\geq0}\frac{q^{n_{1}^2+2n_1n_2+2n_{2}^{2}}}{(q)_{n_1}(q)_{n_{2}}}=\prod_{r\neq 0,\pm 3 \pmod {7}}\frac{1}{1-q^r}=\frac{(q^3;q^7)_\infty (q^4; q^7)_\infty(q^7;q^7)_\infty}{(q)_\infty}\)
  \(A(q)W(q)=AG_{3,3}(q)\)
  where
  \(W(q)=(-q)_{\infty}=\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}\)
  
• Lebesgue's identity
  \(\frac{W(q)^2}{W(q^2)}=\sum_{i,j\geq 0}\frac{q^{(i^2+2ij+2j^2)/2+i/2}}{(q)_{i}(q)_{j}}\)
  

 

(proof)

Note that from useful techniques in q-series

\((-q;q^{2})_{\infty}=\frac{(-q;q)_{\infty}}{(-q^{2};q^{2})_{\infty}}=\frac{(q^{2};q^{2})_{\infty}(q^{2};q^{2})_{\infty}}{(q^{4};q^{4})_{\infty}(q;q)_{\infty}}=\frac{W(q)}{W(q^2)}\)

Therefore

\((-q;q^2)_{\infty}(-q)_{\infty}=\frac{W(q)^2}{W(q^2)}\). ■

 

This corresponds to \(\frac{\eta(\tau)^2}{\eta(2\tau)}\) in the weight 1/2 eta quotients

 

 

history

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

 

 

related items

• #
  

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• 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

• [Alladi&Gordon1993]Partition identities and a continued fraction of Ramanujan
  
  • Krishnaswami Alladi and Basil Gordon, 1993   

• http://www.ams.org/mathscinet
• [1]http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• 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=
• http://ncatlab.org/nlab/show/HomePage

 

 

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/