Bailey pair

"Bailey's method" redirects here, for the root finding algorithm, see Bailey's method (root finding).

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).

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Definition

The q-Pochhammer symbols (a;q)_n are defined as:

(a;q)_n = \prod_{0\le j<n}(1-aq^j) = (1-a)(1-aq)\cdots(1-aq^{n-1}).

A pair of sequences (αnn) 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%2Br}}

or equivalently

\alpha_n = (1-aq^{2n})\sum_{j=0}^n\frac{(aq;q)_{n%2Bj-1}(-1)^{n-j}q^{n-j\choose 2}\beta_j}{(q;q)_{n-j}}.

Bailey's lemma

Bailey's lemma states that if (αnn) is a Bailey pair, then so is (α'n,β'n) where

\alpha^\prime_n= \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^\prime_n = \sum_{j\ge0}\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%2Bn}\sum_{j=-n}^n(-1)^jq^{-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