슬레이터 1

 

 

노트

• 오일러의 오각수정리(pentagonal number theorem)

 

 

항등식의 종류

• 슬레이터 목록 (Slater's list)
• H17

 

 

 

켤레 베일리 쌍의 유도

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

 

 

베일리 쌍의 유도

• 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
  \(a=1,c=0,d=\infty\)
  
• Bailey pair
  \(\alpha_{0}=1\), \(\alpha_{r}=(-1)^{r}(1+q^r)q^{\frac{1}{2}r(r-1)}\)
  \(\beta_{0}=1\), \(\beta_{r}=0\)
  \(\beta_n=\sum_{r=0}^{n}\frac{\alpha_r}{(x)_{n-r}(q)_{n+r}}=\sum_{r=0}^{n}\frac{(-1)^{r}(1+q^r)q^{\frac{1}{2}r(r-1)}}{(q)_{n-r}(q)_{n+r}}=0\)
  

 

 

베일리 쌍

•  
  Bailey pairs
  \(\delta_n=q^{n^2}\)
  \(\gamma_n=\frac{1}{(q)_{\infty}}q^{n^2}\)
   
  \(\alpha_{0}=1\), \(\alpha_{r}=(-1)^{r}(1+q^r)q^{\frac{1}{2}r(r-1)}\)
  \(\beta_{0}=1\), \(\beta_{r}=0\)
  

 

 

q-series 항등식

\(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n=1}^{\infty}(-1)^{n}(q^{\frac{3 n^2-n}{2}}+q^{\frac{3 n^2+n}{2}})\)

• 베일리 쌍(Bailey pair)과 베일리 보조정리
  \(\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}\)
  \(\sum_{n=0}^{\infty}\beta_n\delta_{n}=1\)
  \(\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\frac{1}{(q)_{\infty}}(1+\sum_{n=1}^{\infty}(-1)^{n}(q^{\frac{3 n^2-n}{2}}+q^{\frac{3 n^2+n}{2}}))\)
  

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