"Bailey pair and lemma"의 두 판 사이의 차이

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<h5 style="MARGIN: 0px; LINE-HEIGHT: 2em;">Bailey chain</h5>
 
<h5 style="MARGIN: 0px; LINE-HEIGHT: 2em;">Bailey chain</h5>
  
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* [[search?q=Bailey%20chain&parent id=5562297|Bailey chain]]<br>
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* [[6080259|Bailey chain]]<br>
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*  we derive a new Bailey chain from a known Bailey pair<br><math>\alpha^\prime_n= \frac{(\rho_1;q)_n(\rho_2;q)_n(aq/\rho_1\rho_2)^n}{(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\alpha_n</math><br><math>\beta^\prime_n = \sum_{j\ge0}\frac{(\rho_1;q)_j(\rho_2;q)_j(aq/\rho_1\rho_2;q)_{n-j}(aq/\rho_1\rho_2)^j}{(q;q)_{n-j}(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\beta_j</math><br>
 
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*  corollary. by taking  <math>\rho_1,\rho_2\to \infty</math> , we get <br><math>\alpha^\prime_n= a^nq^{n^2}\alpha_n</math><br><math>\beta^\prime_n = \sum_{r=0}^{L}\frac{a^rq^{r^2}}{(q)_{L-r}}\beta_j</math><br>
 
  
 
 
 
 
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* [http://dx.doi.org/10.1112/plms/s2-53.6.460 A New Proof of Rogers's Transformations of Infinite Series]<br>
 
* [http://dx.doi.org/10.1112/plms/s2-53.6.460 A New Proof of Rogers's Transformations of Infinite Series]<br>
 
**  Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475<br>
 
**  Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475<br>
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*  On two theorems of combinatory analysis and some allied identities <br>
 
*  On two theorems of combinatory analysis and some allied identities <br>
 
* http://www.ams.org/mathscinet<br>
 
* http://www.ams.org/mathscinet<br>

2010년 7월 14일 (수) 21:37 판

introduction
  •  q-Pfaff-Sallschutz sum

 

 

Bailey lemma

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,

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

 

(corollay 1)

Choose the following

\(u_{n}=\frac{1}{(q)_n}\) ,\(v_{n}=\frac{1}{(x)_n}\),\(\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}\)

Then 

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

and hence by Bailey's lemma,

\(\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\frac{(x/y;q)_{\infty}(x/z;q)_{\infty}}{(x;q)_{\infty}(x/yz;q)_{\infty}}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}\)

(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 \(a=yq^{n},b=zq^{n},c=xq^{2n}\). 

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

 

 

 

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  \(\{\delta_r\}, \{\gamma_r\}\)
    \(\gamma_L=\sum_{r=L}^{\infty}\frac{\delta_r}{(q)_{r-L}(aq)_{r+L}}\)

 

 

 

Bailey chain

 

 

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