"Bailey pair and lemma"의 두 판 사이의 차이

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38번째 줄: 38번째 줄:
  
 
*  When we have a Bailey pair, we can produce q-series identities<br>
 
*  When we have a Bailey pair, we can produce q-series identities<br>
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*  the Bailey lemma gives an identity involving q-series<br>
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**  (1) Bailey lemma gives an identity involving q-series<br>
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*  using the definition of Bailey pair<br><math>\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}</math><br>
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**  (2) using the definition of Bailey pair<br><math>\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}</math><br>
  
 
 
 
 
47번째 줄: 47번째 줄:
 
<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 2em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">Bailey lemma</h5>
 
<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 2em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">Bailey lemma</h5>
  
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If the sequence <math>\{\alpha_r\}, \{\beta_r\}</math>, <math>\{\delta_r\}, \{\gamma_r\}</math> satisfy the following
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*  Bailey lemma involved a Bailey pair and a conjugate Bailey pair<br>
  
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<math>\beta_L=\sum_{r=0}^{L}{\alpha_r}{u_{L-r}v_{L+r}}</math>, <math>\gamma_L=\sum_{r=L}^{\infty}{\delta_r}{u_{r-L}v_{r+L}}</math>
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*  If the sequence <math>\{\alpha_r\}, \{\beta_r\}</math>, <math>\{\delta_r\}, \{\gamma_r\}</math> satisfy the following<br><math>\beta_L=\sum_{r=0}^{L}{\alpha_r}{u_{L-r}v_{L+r}}</math>, <math>\gamma_L=\sum_{r=L}^{\infty}{\delta_r}{u_{r-L}v_{r+L}}</math><br> then,<br><math>\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}</math><br>
  
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then,
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<math>\sum_{n=0}^{\infty}\alpha_n\gamma_{n}=\sum_{n=0}^{\infty}\beta_n\delta_{n}</math>
 
  
 
 
 
 
  
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(corolary 1)
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<h5 style="margin: 0px; line-height: 2em;">specialization</h5>
  
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Choose the following (in the following, x=aq to get a Bailey pair relative to a)
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*  Choose the following (in the following, x=aq to get a Bailey pair relative to a)<br><math>u_{n}=\frac{1}{(q)_n}</math> ,<math>v_{n}=\frac{1}{(x)_n}</math>,<br>
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*  There is a conjugate Bailey pair<br><math>\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}</math><br>  <math>\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}</math>,<br>
  
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<math>u_{n}=\frac{1}{(q)_n}</math> ,<math>v_{n}=\frac{1}{(x)_n}</math>,<math>\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}</math>
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Then 
 
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 <math>\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}</math>,
 
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and hence by Bailey's lemma,
 
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<math>\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}</math>
 
  
 
(proof)
 
(proof)
81번째 줄: 72번째 줄:
 
Put <math>a=yq^{n},b=zq^{n},c=xq^{2n}</math>. 
 
Put <math>a=yq^{n},b=zq^{n},c=xq^{2n}</math>. 
  
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<math>\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}</math> (a different notation <math>\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}</math> is also used sometimes) ■
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<math>\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}</math>
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<math>\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}</math> 
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(a different notation
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 +
 <math>\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}</math>
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 +
is also used sometimes) ■
 +
 
 +
*  If we apply Bailey lemma to the above conjugate pair, we get<br><math>\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}</math><br>
  
 
 
 
 

2010년 9월 17일 (금) 21:43 판

introduction
  •  q-Pfaff-Sallschutz sum

 

 

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  \(\{\delta_r\}, \{\gamma_r\}\)
    \(\gamma_L=\sum_{r=L}^{\infty}\frac{\delta_r}{(q)_{r-L}(aq)_{r+L}}\)
  • Note that the conjugate Bailey pair is different from a Bailey pair

 

 

 

examples of Bailey pair

 

 

how to obtain and check Bailey pair?
  • various complicated q-series identities

 

 

why do we care about Bailey pair?
  • When we have a Bailey pair, we can produce q-series identities
    • (1) Bailey lemma gives an identity involving q-series
    • (2) using the definition of Bailey pair
      \(\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}\)

 

 

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 

(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 \(a=yq^{n},b=zq^{n},c=xq^{2n}\). 

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

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

(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

 

 

 

Bailey chain

 

 

history

 

 

 

related items

 

 

encyclopedia

 

 

books

[[4909919|]]

 

 

articles

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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