Slater 83

Note

• Slater 86
  

 

 

type of identity

• Slater's list
•  
  

 

 

Bailey pair 1

• Use the folloing
  \(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\),  \(\gamma_n=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\)
  
• Specialize
  \(x=q^2, y=-q, z\to\infty\).
  
• Bailey pair
  \(\delta_n=(-q)_{n}q^{\frac{n(n+1)}{2}}\)
  \(\gamma_n=\frac{(-q)_{\infty}}{(q^2)_{\infty}}q^{\frac{n(n+1)}{2}}\)
  

 

 

 

Bailey pair 2

• Use the following 
  \(\sum_{r=0}^{n}\frac{(1-aq^{2r})(-1)^{r}q^{\frac{1}{2}(r^2+r)}(a)_{r}(c)_{r}(d)_{r}a^{r}}{(a)_{n+r+1}(q)_{n-r}(q)_{r}(aq/c)_{r}(aq/d)_{r}c^{r}d^{r}}=\frac{(aq/cd)_{n}}{(q)_{n}(aq/c)_{n}(aq/d)_{n}}\)
  
• Specialize
  \(a=q,c=-q,d=\infty\)
  
• Bailey pair
  \(\alpha_{0}=1\), \(\alpha_{n}=(-1)^{n}q^{n^2}(1-q^{2n+1})/(1-q)\)
  \(\beta_n=\sum_{r=0}^{n}\frac{\alpha_r}{(x)_{n-r}(q)_{n+r}}=\sum_{r=0}^{n}\frac{\alpha_r}{(q^{2})_{n-r}(q)_{n+r}}=\frac{1}{(q)_{n}(-q)_{n}}\)
  

 

 

Bailey pair 

• Bailey pairs
  \(\delta_n=(-q)_{n}q^{\frac{n(n+1)}{2}}\)
  \(\gamma_n=\frac{(-q)_{\infty}}{(q^2)_{\infty}}q^{\frac{n(n+1)}{2}}\)
  \(\alpha_{n}=(-1)^{n}q^{n^2}(1-q^{2n+1})/(1-q)\)
  \(\beta_n=\frac{1}{(q)_{n}(-q)_{n}}\)
  

 

 

 

q-series identity

\(\sum_{n=0}^{\infty}\frac{q^{2n^{2}}}{ (q)_{2n}}=\frac{(q^{1};q^{8})_{\infty}(q^{7};q^{8})_{\infty}(q^{8};q^{8})_{\infty}(q^{6};q^{16})_{\infty}(q^{10};q^{16})_{\infty}}{(q)_{\infty}}\)

\((q)_{2n}=(q;q^2)_{n}(q^2;q^2)_{n}\)

• Bailey's lemma
  \(\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}\)
  \(\sum_{n=0}^{\infty}\beta_n\delta_{n}=\sum_{n=0}^{\infty}\frac{q^{\frac{n(n+1)}{2}}}{(q)_{n}}\)
  

• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

Bethe type equation (cyclotomic equation)

Let 
\(\sum_{n=0}^{\infty}\frac{q^{n(an+b)/2}}{ \prod_{j=1}^{r}(q^{c_j};q^{d_j})_n^{e_j}}=\sum_{N=0}^{\infty} a_N q^{N}\).

Then \(\prod_{j=1}^{r}(1-x^{d_j})^{e_j}=x^a\)  has a unique root \(0<\mu<1\). We get

\(\log^2 a_N \sim 4N\sum_{j=1}^{r}\frac{e_j}{d_j}L(1-\mu^{d_j})\)

 

a=4,d_1=2.d_2=2,e=1

The equation  becomes \((1-x^{2})^{2}=x^{4}\).

\(x^2=\frac{1}{2}\)

\(4L(\frac{1}{2})=\frac{1}{2}(\frac{2}{3}\pi^2)=\frac{1}{3}\pi^2\)

 

 

dilogarithm identity

\(L(\frac{1}{2})=\frac{1}{12}\pi^2\)

 

 

articles

• Hypergeometric generating function of $L$-function, Slater's identities, and quantum invariant
  
  • Kazuhiro Hikami, Anatol N. Kirillov, 2004