"Bailey pair and lemma"의 두 판 사이의 차이

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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">introduction</h5>
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==related items==
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* [[manufacturing matrices from lower ranks]]
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* [[q-analogue of summation formulas]]
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* [[Rogers-Ramanujan continued fraction]]
  
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*  q-Pfaff-Sallschutz sum<br>
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==articles==
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* Patkowski, Alexander E. ‘A Note on Some Partitions Related to Ternary Quadratic Forms’. arXiv:1503.08516 [math], 29 March 2015. http://arxiv.org/abs/1503.08516.
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*  A. Schilling, S.O. Warnaar [http://arxiv.org/abs/math.QA/9909044 A generalization of the q-Saalschutz sum and the Burge transform], 2009
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* Mc Laughlin [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities], 2008
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* [http://www.springerlink.com/content/478544t414l26v05/ Andrews–Gordon type identities from combinations of Virasoro characters]
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**  Boris Feigin, Omar Foda, Trevor Welsh, 2007
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* [http://www.combinatorics.org/Volume_10/PDF/v10i1r13.pdf Finite Rogers-Ramanujan Type Identities]
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**  Andrew V. Sills, 2003
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* [http://dx.doi.org/10.1142/S0217751X97001110 Virasoro character identities from the Andrews–Bailey construction]
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**  Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)
  
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<h5 style="margin: 0px; line-height: 2em;">Bailey pair</h5>
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* [http://projecteuclid.org/euclid.pjm/1102708707 Multiple series Rogers-Ramanujan type identities.]
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** George E. Andrews, Pacific J. Math. Volume 114, Number 2 (1984), 267-283.
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*  the sequence <math>\{\alpha_r\}, \{\beta_r\}</math> satisfying the following is called a Bailey pair relative to a<br><math>\beta_L=\sum_{r=0}^{L}\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}</math><br>
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* [http://matwbn.icm.edu.pl/ksiazki/aa/aa43/aa4326.pdf Special values of the dilogarithm function]
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*  conjugate Bailey pair  <math>\{\delta_r\}, \{\gamma_r\}</math><br><math>\gamma_L=\sum_{r=L}^{\infty}\frac{\delta_r}{(q)_{r-L}(aq)_{r+L}}=\sum_{r=0}^{\infty}\frac{\delta_{r+L}}{(q)_{r}(aq)_{r+2L}}</math><br>
 
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*  Note that the summation for a conjugate Bailey pair is different from that of a Bailey pair<br>
 
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<h5 style="margin: 0px; line-height: 2em;">examples of Bailey pair</h5>
 
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* [[Slater list|Slater's list]]<br>
 
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<h5 style="margin: 0px; line-height: 2em;">how to obtain and check Bailey pair?</h5>
 
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*  various complicated q-series identities<br>
 
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<h5 style="margin: 0px; line-height: 2em;">why do we care about Bailey pair?</h5>
 
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*  When we have a Bailey pair, we can produce q-series identities<br>
 
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**  (1) Bailey lemma gives an identity involving q-series<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>
 
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<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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*  Bailey lemma involved a Bailey pair and a conjugate Bailey pair<br>
 
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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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<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)<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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(proof)
 
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By the basic analogue of Gauss' theorem 
 
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<math>\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}}</math>,
 
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See
 
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[[q-analogue of summation formulas|]]
 
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* [http://pythagoras0.springnote.com/pages/9408516 합공식의 q-analogue]
 
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Also note that <math>(a)_{n+r}=(a)_{r}(aq^{r})_{n}</math> and <math>(a)_{\infty}=(a)_{r}(aq^{r})_{\infty}</math>
 
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Put <math>a=yq^{r},b=zq^{r},c=xq^{2r}</math>. 
 
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Then we get (*)
 
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<math>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</math>
 
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From the left hand side,
 
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<math>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}</math>
 
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Now let
 
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<math>\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}</math> so that
 
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<math>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}}</math>
 
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From the right hand side of (*), we get
 
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<math>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}}</math>
 
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Therefore,
 
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<math>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</math>
 
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By simplifying the above equation, we obtain
 
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<math>\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}}</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>  with <math>\delta_n=\frac{(y)_n(z)_n x^n}{y^n z^n}</math> gives a conjugate Bailey pair
 
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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) ■
 
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*  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>
 
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<h5 style="margin: 0px; line-height: 2em;">examples</h5>
 
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*  Conjugate Bailey pair (<math>x=q,y\to\infty, z\to\infty</math>)<br><math>\delta_n=q^{n^2}</math><br><math>\gamma_n=\frac{q^{n^2}}{(q)_{\infty}}</math><br>
 
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*  Bailey pair<br><math>\alpha_{n}=(-1)^{n}q^{\frac{3}{2}n^2}(q^{\frac{1}{2}n}+q^{-\frac{1}{2}n})</math><br><math>\beta_n=\frac{1}{(q)_{n}}</math><br>
 
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*  we get the Rogers-Ramanujan identity([[5974537|Slater 18]])<br><math>\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}}</math><br>
 
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<h5 style="margin: 0px; line-height: 2em;">Bailey chain</h5>
 
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* [[6080259|Bailey chain]]<br>
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">related items</h5>
 
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* [[Bloch group]]<br>
 
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* [[Bloch group, K-theory and dilogarithm]]<br>
 
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* [[1 manufacturing matrices from lower ranks|manufacturing matrices from lower ranks]]<br>
 
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* [[q-analogue of summation formulas]]<br>
 
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* [[Rogers-Ramanujan continued fraction]]<br>
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">encyclopedia</h5>
 
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* http://en.wikipedia.org/wiki/Bailey_pair
 
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* http://en.wikipedia.org/wiki/Wilfrid_Norman_Bailey
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">books</h5>
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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[[4909919|]]
 
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<h5 style="margin: 0px; line-height: 2em;">expositions</h5>
 
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* [http://arxiv.org/abs/0910.2062v2 50 Years of Bailey's lemma] S. Ole Warnaar, 2009<br>
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">articles</h5>
 
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* [http://arxiv.org/abs/math.QA/9909044 A generalization of the q-Saalschutz sum and the Burge transform]<br>
 
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**  A. Schilling, S.O. Warnaar, 2009<br>
 
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* [http://www.combinatorics.org/Surveys/ds15.pdf Rogers-Ramanujan-Slater Type identities]<br>
 
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**  Mc Laughlin, 2008<br>
 
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* [http://www.springerlink.com/content/478544t414l26v05/ Andrews–Gordon type identities from combinations of Virasoro characters]<br>
 
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**  Boris Feigin, Omar Foda, Trevor Welsh, 2007<br>
 
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* [http://www.combinatorics.org/Volume_10/PDF/v10i1r13.pdf Finite Rogers-Ramanujan Type Identities]<br>
 
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**  Andrew V. Sills, 2003<br>
 
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* [http://dx.doi.org/10.1142/S0217751X97001110 Virasoro character identities from the Andrews–Bailey construction]<br>
 
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**  Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)<br>
 
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* [http://projecteuclid.org/euclid.pjm/1102708707 Multiple series Rogers-Ramanujan type identities.]<br>
 
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** George E. Andrews, Pacific J. Math.  Volume 114, Number 2 (1984), 267-283.<br>
 
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* [http://matwbn.icm.edu.pl/ksiazki/aa/aa43/aa4326.pdf Special values of the dilogarithm function]<br>
 
 
** J. H. Loxton, 1984
 
** J. H. Loxton, 1984
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* [http://dx.doi.org/10.1112%2Fjlms%2Fs1-37.1.504 Wilfrid Norman Bailey]<br>
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* [http://dx.doi.org/10.1112%2Fjlms%2Fs1-37.1.504 Wilfrid Norman Bailey]
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**  Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512<br>
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**  Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512
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* [http://dx.doi.org/10.1112%2Fplms%2Fs2-54.2.147 Further identities of the Rogers-Ramanujan type]<br>
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* [http://dx.doi.org/10.1112%2Fplms%2Fs2-54.2.147 Further identities of the Rogers-Ramanujan type]
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**  Slater, L. J. (1952),  Proceedings of the London Mathematical Society. Second Series 54: 147–167<br>
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**  Slater, L. J. (1952), Proceedings of the London Mathematical Society. Second Series 54: 147–167
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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>
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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]
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**  Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475<br>
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**  Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475
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* [http://plms.oxfordjournals.org/cgi/reprint/s2-50/1/1.pdf Identities of Rogers-Ramanujan type]<br>
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* [http://plms.oxfordjournals.org/cgi/reprint/s2-50/1/1.pdf Identities of Rogers-Ramanujan type]
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**  Bailey, 1944<br>
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**  Bailey, 1944
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*  On two theorems of combinatory analysis and some allied identities <br>
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[[분류:math and physics]]
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* http://www.ams.org/mathscinet<br>
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[[분류:migrate]]
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html][http://www.ams.org/mathscinet ]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://dx.doi.org/10.1112/plms/s2-53.6.460
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">blogs</h5>
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">links</h5>
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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==메타데이터==
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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===위키데이터===
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* ID :  [https://www.wikidata.org/wiki/Q4848398 Q4848398]
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* http://functions.wolfram.com/
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===Spacy 패턴 목록===
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*
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* [{'LOWER': 'bailey'}, {'LEMMA': 'pair'}]

2021년 2월 17일 (수) 01:50 기준 최신판

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Spacy 패턴 목록

  • [{'LOWER': 'bailey'}, {'LEMMA': 'pair'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Bailey_pair_and_lemma&oldid=51844"