Bailey pair and lemma

 

 

 

examples of Bailey pair

• Slater's list
  

 

 

Bailey lemma

• Bailey lemma involved a Bailey pair and a conjugate Bailey pair
  

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

 

 

specialization

• Choose the following (in the following, x=aq to get a Bailey pair relative to a)
  \(u_{n}=\frac{1}{(q)_n}\) ,\(v_{n}=\frac{1}{(x)_n}\),
  
• There is a conjugate Bailey pair
  \(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\)
   \(\gamma_n=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\)
  

 

(proof)

By the basic analogue of Gauss' theorem 

\(\sum_{n=0}^{\infty}\frac{(a,q)_{n}(b,q)_{n}}{(c ,q)_{n}(q ,q)_{n}}(\frac{c}{ab})^{n}=\frac{(c/a;q)_{\infty}(c/b;q)_{\infty}}{(c;q)_{\infty}(c/(ab);q)_{\infty}}\),

[[q-analogue of summation formulas|]]

• 합공식의 q-analogue
• q-가우스 합

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

Put \(a=yq^{r},b=zq^{r},c=xq^{2r}\). 

Then we get (*)

\(A=\sum_{n=0}^{\infty}\frac{(yq^{r})_{n}(zq^{r})_{n}}{(xq^{2r})_{n}(q)_{n}}(\frac{x}{yz})^{n}=\frac{(xq^{r}/y)_{\infty}(xq^{r}/z)_{\infty}}{(xq^{2r})_{\infty}(x/(yz))_{\infty}}=B\)

From the left hand side,

\(A=\sum_{n=0}^{\infty}\frac{(yq^{r})_{n}(zq^{r})_{n}}{(xq^{2r})_{n}(q)_{n}}(\frac{x}{yz})^{n}=\frac{(x)_{2r}}{(y)_{r}(z)_{r}}\sum_{n=0}^{\infty}\frac{(y)_{n+r}(z)_{n+r}}{(x)_{n+2r}(q)_{n}}(\frac{x}{yz})^{n}=\frac{(x)_{2r}y^{r}z^{r}}{(y)_{r}(z)_{r}x^{r}}\sum_{n=0}^{\infty}\frac{(y)_{n+r}(z)_{n+r}}{(x)_{n+2r}(q)_{n}}(\frac{x}{yz})^{n+r}\)

Now let

\(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\) so that

\(A=\frac{(x)_{2r}y^{r}z^{r}}{(y)_{r}(z)_{r}x^{r}}\sum_{n=0}^{\infty}\frac{\delta_{n+r}}{(x)_{n+2r}(q)_{n}}\)

From the right hand side of (*), we get

\(B=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(x)_{2r}}{(x/y)_{r}(x/z)_{r}}\)

Therefore,

\(A=\frac{(x)_{2r}y^{r}z^{r}}{(y)_{r}(z)_{r}x^{r}}\sum_{n=0}^{\infty}\frac{\delta_{n+r}}{(x)_{n+2r}(q)_{n}}=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(x)_{2r}}{(x/y)_{r}(x/z)_{r}}=B\)

By simplifying the above equation, we obtain

\(\sum_{n=0}^{\infty}\frac{\delta_{n+r}}{(x)_{n+2r}(q)_{n}}=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{1}{(x/y)_{r}(x/z)_{r}}\frac{(y)_{r}(z)_{r}x^{r}}{y^{r}z^{r}}\)

\(\gamma_n=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\)  with \(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\) gives a conjugate Bailey pair

(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) ■

• If we apply Bailey lemma to the above conjugate pair, we get
  \(\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}\)
  

 

 

examples

• Conjugate Bailey pair (\(x=q,y\to\infty, z\to\infty\))
  \(\delta_n=q^{n^2}\)
  \(\gamma_n=\frac{q^{n^2}}{(q)_{\infty}}\)
  
• Bailey pair
  \(\alpha_{n}=(-1)^{n}q^{\frac{3}{2}n^2}(q^{\frac{1}{2}n}+q^{-\frac{1}{2}n})\)
  \(\beta_n=\frac{1}{(q)_{n}}\)
  
• we get the Rogers-Ramanujan identity(Slater 18)
  \(\sum_{n=0}^{\infty}\frac{q^{n^2}}{ (q)_{n}}=\frac{(q^{3};q^{5})_{\infty}(q^{2};q^{5})_{\infty}(q^{5};q^{5})_{\infty}}{(q)_{\infty}}=\frac{1}{(q^{1};q^{5})_{\infty}(q^{4};q^{5})_{\infty}}\)
  

 

 

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
  
• q-analogue of summation formulas
  
• Rogers-Ramanujan continued fraction
  

 

 

articles

•  
  A. Schilling, S.O. Warnaar A generalization of the q-Saalschutz sum and the Burge transform, 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