Bailey pair
In mathematics, a Bailey pair is a pair of sequences satisfying certain relations, and a Bailey chain is a sequence of Bailey pairs. Bailey pairs were introduced by W. N. Bailey (1947, 1948) while studying the second proof Rogers (1917) of the Rogers–Ramanujan identities, and Bailey chains were introduced by Andrews (1984).
Definition
The q-Pochhammer symbols are defined as:
A pair of sequences (αn,βn) is called a Bailey pair if they are related by
or equivalently
Bailey's lemma
Bailey's lemma states that if (αn,βn) is a Bailey pair, then so is (α'n,β'n) where
In other words, given one Bailey pair, one can construct a second using the formulas above. This process can be iterated to produce an infinite sequence of Bailey pairs, called a Bailey chain.
Examples
An example of a Bailey pair is given by (Andrews, Askey & Roy 1999, p. 590)
L. J. Slater (1952) gave a list of 130 examples related to Bailey pairs.
References
- Andrews, George E. (1984), "Multiple series Rogers-Ramanujan type identities", Pacific Journal of Mathematics, 114 (2): 267–283, doi:10.2140/pjm.1984.114.267, ISSN 0030-8730, MR 757501
- Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, ISBN 978-0-521-62321-6, MR 1688958
- Bailey, W. N. (1947), "Some identities in combinatory analysis", Proceedings of the London Mathematical Society, Second series, 49 (6): 421–425, doi:10.1112/plms/s2-49.6.421, ISSN 0024-6115, MR 0022816
- Bailey, W. N. (1948), "Identities of the Rogers-Ramanujan Type", Proc. London Math. Soc., s2-50 (1): 1–10, doi:10.1112/plms/s2-50.1.1
- Paule, Peter, The Concept of Bailey Chains (PDF)
- Slater, L. J. (1952), "Further identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society, Second series, 54 (2): 147–167, doi:10.1112/plms/s2-54.2.147, ISSN 0024-6115, MR 0049225
- Warnaar, S. Ole (2001), "50 years of Bailey's lemma", Algebraic combinatorics and applications (Gössweinstein, 1999) (PDF), Berlin, New York: Springer-Verlag, pp. 333–347, MR 1851961