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 $(a;q)_{n}$ are defined as:

(a;q)_{n}=\prod _{0\leq j<n}(1-aq^{j})=(1-a)(1-aq)\cdots (1-aq^{n-1}).

A pair of sequences (α_n,β_n) is called a Bailey pair if they are related by

\beta _{n}=\sum _{r=0}^{n}{\frac {\alpha _{r}}{(q;q)_{n-r}(aq;q)_{n+r}}}

or equivalently

\alpha _{n}=(1-aq^{2n})\sum _{j=0}^{n}{\frac {(aq;q)_{n+j-1}(-1)^{n-j}q^{n-j \choose 2}\beta _{j}}{(q;q)_{n-j}}}.

Bailey's lemma

Bailey's lemma states that if (α_n,β_n) is a Bailey pair, then so is (α'_n,β'_n) where

\alpha _{n}^{\prime }={\frac {(\rho _{1};q)_{n}(\rho _{2};q)_{n}(aq/\rho _{1}\rho _{2})^{n}\alpha _{n}}{(aq/\rho _{1};q)_{n}(aq/\rho _{2};q)_{n}}}

\beta _{n}^{\prime }=\sum _{j\geq 0}{\frac {(\rho _{1};q)_{j}(\rho _{2};q)_{j}(aq/\rho _{1}\rho _{2};q)_{n-j}(aq/\rho _{1}\rho _{2})^{j}\beta _{j}}{(q;q)_{n-j}(aq/\rho _{1};q)_{n}(aq/\rho _{2};q)_{n}}}.

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)

\alpha _{n}=q^{n^{2}+n}\sum _{j=-n}^{n}(-1)^{j}q^{-j^{2}},\quad \beta _{n}={\frac {(-q)^{n}}{(q^{2};q^{2})_{n}}}.

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 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