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 while studying the second proof Rogers 1917 of the Rogers–Ramanujan identities, and Bailey chains were introduced by.

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 (αn,βn) 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 (αn,βn) 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
 * $$\alpha_n = q^{n^2+n}\sum_{j=-n}^n(-1)^jq^{-j^2}, \quad \beta_n = \frac{(-q)^n}{(q^2;q^2)_n}. $$

gave a list of 130 examples related to Bailey pairs.