"Bailey pair and lemma"의 두 판 사이의 차이
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| 33번째 줄: | 33번째 줄: | ||
Hence by Bailey's lemma, | Hence by Bailey's lemma, | ||
| − | <math>\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}</math> | + | <math>\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}</math> |
| 131번째 줄: | 131번째 줄: | ||
* [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> | ||
| − | * On two theorems | + | * On two theorems of combinatory analysis and some allied identities <br> |
* http://www.ams.org/mathscinet<br> | * http://www.ams.org/mathscinet<br> | ||
2010년 6월 23일 (수) 18:35 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)
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 by the basic analogue of Gauss's theorem (q-Gauss sum)
\(\gamma_n=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\)
Hence by Bailey's lemma,
\(\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{y^n z^n}\beta_{n}=\prod{{x/y,x/z;q}\choose {x,x/yz;}}\sum_{n=0}^{\infty}\frac{(y)_n(z)_n x^n}{(x/y)_{n}(x/z)_{n}y^n z^n}\alpha_{n}\)
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
- we derive a new Bailey chain from a known Bailey pair
\(\alpha^\prime_n= \frac{(\rho_1;q)_n(\rho_2;q)_n(aq/\rho_1\rho_2)^n}{(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\alpha_n\)
\(\beta^\prime_n = \sum_{j\ge0}\frac{(\rho_1;q)_j(\rho_2;q)_j(aq/\rho_1\rho_2;q)_{n-j}(aq/\rho_1\rho_2)^j}{(q;q)_{n-j}(aq/\rho_1;q)_n(aq/\rho_2;q)_n}\beta_j\) - corollary. by taking \(\rho_1,\rho_2\to \infty\) , we get
\(\alpha^\prime_n= a^nq^{n^2}\alpha_n\)
\(\beta^\prime_n = \sum_{r=0}^{L}\frac{a^rq^{r^2}}{(q)_{L-r}}\beta_j\)
history
encyclopedia
- http://en.wikipedia.org/wiki/Bailey_pair
- http://en.wikipedia.org/wiki/Wilfrid_Norman_Bailey
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
[[4909919|]]
articles
- 50 Years of Bailey's lemma
- S. Ole Warnaar, 2009
- S. Ole Warnaar, 2009
- A generalization of the q-Saalschutz sum and the Burge transform
- A. Schilling, S.O. Warnaa, 2009
- A. Schilling, S.O. Warnaa, 2009
- Rogers-Ramanujan-Slater Type identities
- Mc Laughlin, 2008
- Mc Laughlin, 2008
- Andrews–Gordon type identities from combinations of Virasoro characters
- Boris Feigin, Omar Foda, Trevor Welsh, 2007
- Boris Feigin, Omar Foda, Trevor Welsh, 2007
- Finite Rogers-Ramanujan Type Identities
- Andrew V. Sills, 2003
- Andrew V. Sills, 2003
- Virasoro character identities from the Andrews–Bailey construction
- Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)
- Foda, O., Quano, Y.-H, Int. J. Mod. Phys. A 12, 1651–1675 (1997)
- Multiple series Rogers-Ramanujan type identities.
- George E. Andrews, Pacific J. Math. Volume 114, Number 2 (1984), 267-283.
- George E. Andrews, Pacific J. Math. Volume 114, Number 2 (1984), 267-283.
- Special values of the dilogarithm function
- J. H. Loxton, 1984
- Wilfrid Norman Bailey
- Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512
- Slater, L. J. (1962), Journal of the London Mathematical Society. Second Series 37: 504–512
- Further identities of the Rogers-Ramanujan type
- Slater, L. J. (1952), Proceedings of the London Mathematical Society. Second Series 54: 147–167
- Slater, L. J. (1952), Proceedings of the London Mathematical Society. Second Series 54: 147–167
- A New Proof of Rogers's Transformations of Infinite Series
- Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475
- Slater, L. J. (1952), Proc. London Math. Soc. 1951 s2-53: 460-475
- On two theorems of combinatory analysis and some allied identities
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html[1]
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- 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=
- http://dx.doi.org/10.1112/plms/s2-53.6.460
question and answers(Math Overflow)
blogs
experts on the field