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

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The q-Pochhammer symbols   are defined as:

 

A pair of sequences (αnn) is called a Bailey pair if they are related by

 

or equivalently

 

Bailey's lemma

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Bailey's lemma states that if (αnn) 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

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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

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  • 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 0757501
  • Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, vol. 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