Bailey pair explained

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} ( 1- aq ^{j)} = ( 1- a)( 1- aq) … ( 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} ^{n (\alpha _{r})/((q;q) _{n-r} (aq;q) _{n+r})} ^{r =0} or equivalently\alpha _{n} = ( 1- aq ^{2n} )\sum ^{n ((aq;q) _{n+j-1} ( -1 ) ^{n-j} q ^{n-j\choose} \beta _{j})/((q;q) _{n-j})} _{j =0} .

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 ^{\prime} _{n =} ((\rho _{1 ;q)} _{n(\rho} _{2;q)} _{n(aq/\rho} _{1\rho} ^{n\alpha} ^{n})/((aq/\rho _{1;q)} _{n(aq/\rho} _{2;q)} _{n}) \beta ^{\prime} _{n} = \sum _{j\ge0} ((\rho _{1 ;q)} _{j(\rho} _{2;q)} _{j(aq/\rho} _{1\rho} _{2;q)} _{n-j} (aq/\rho _{1\rho} ^{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 ^{n( -1 )} _{j =- n} ^{jq} ^{- j ^{2}} , \beta _{n} = (( - q) ^{n})/((q ^{2 ;q} ^{2 )} ^{n}) .

gave a list of 130 examples related to Bailey pairs.