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 \(aq^{-1}\)

corollary

Let \(\{\alpha_r\}, \{\beta_r\}\) be the initial Bailey pair relative to a

apply Bailey chain construction k-i-1 times

At the (k-i)th step apply Bailey lattice

apply Bailey 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 \(\{\alpha_r'\}, \{\beta_r'\}\),

\(\beta_L'=\sum_{r=0}^{L}\frac{\alpha_r'}{(q)_{L-r}(q)_{L+r}}\)

Take L to limit.

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related items

encyclopedia

http://en.wikipedia.org/wiki/
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Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

[[4909919|]]

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.
The Bailey Lattice
- A. Agarwal, G.E. Andrews, and D. Bressoud, J. Indian Math. Soc. 51 (1987), 57-73.
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