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) harvtxt error: no target: CITEREFRogers1917 (help) of the Rogers–Ramanujan identities, and Bailey chains were introduced by Andrews (1984).

Definition

The q-Pochhammer symbols ${\displaystyle (a;q)_{n}}$ are defined as:

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

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

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

or equivalently

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

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

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

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

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)

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

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, 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