Bailey pair

Definition

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

:(a;q)n = \prod{0\le j

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.

References