Bailey lattice
introduction
Let \(\{\alpha_r\}, \{\beta_r\}\) be a Bailey pair relative to a and set
\(\alpha_0'=0\), \(\alpha_n'=(1-a)a^nq^{n^2-n}(\frac{\alpha_n}{1-aq^{2n}}-\frac{aq^{2n-2}\alpha_{n-1}}{1-aq^{2n-2}})\)\(\beta_n'=\sum_{r=0}^{n}\frac{a^rq^{r^2-r}}{(q)_{n-r}}\beta_{r}\)
Then \(\{\alpha_r'\}, \{\beta_r'\}\) is a Bailey pair relative to \(aq^{-1}\)
comparison with Bailey chain
- Bailey chain
\(\alpha^\prime_n= a^nq^{n^2}\alpha_n\)
\(\beta^\prime_L = \sum_{r=0}^{L}\frac{a^rq^{r^2}}{(q)_{L-r}}\beta_r\) - This does not change the parameter a of the Bailey pair.
- lattice construction changes this
corollary
Let \(\{\alpha_r\}, \{\beta_r\}\) be the initial Bailey pair relative to a
apply Bailey chain construction k-i-1 times (Bailey chain)
At the (k-i)th step apply Bailey lattice
apply Bailey chain construction i times again.
Then we get a Bailey pair
\(\{\alpha_r'\}, \{\beta_r'\}\) is a Bailey pair relative to \(aq^{-1}\).
If we use the defining relation of Bailey pair to \(\{\alpha_r'\}, \{\beta_r'\}\),
\(\beta_L'=\sum_{r=0}^{L}\frac{\alpha_r'}{(q)_{L-r}(q)_{L+r}}\)
and take the limit L\to\infty
history
encyclopedia
- 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=
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articles
- A Bailey Lattice
- Jeremy Lovejoy, Proceedings of the American Mathematical Society, Vol. 132, No. 5 (May, 2004), pp. 1507-1516
- The Bailey lattice
- David Bressoud, an introduction, pp. 57--67 in Ramanujan Revisited. G. E. Andrews et al. eds., Academic Press, 1988.
- David Bressoud, an introduction, pp. 57--67 in Ramanujan Revisited. G. E. Andrews et al. eds., Academic Press, 1988.
- The Bailey Lattice
- A. Agarwal, G.E. Andrews, and D. Bressoud, J. Indian Math. Soc. 51 (1987), 57-73.
- A. Agarwal, G.E. Andrews, and D. Bressoud, J. Indian Math. Soc. 51 (1987), 57-73.
- http://www.ams.org/mathscinet
- [1]http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/line-height: 2em;">
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
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experts on the field