Here's an excerpt of what Wikipedia says https://en.wikipedia.org/wiki/Bailey_pair
A pair of sequences $(α_n,β_n)$ 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 states that if $(α_n,β_n)$ is a Bailey pair, then so is $(α'_n,β'_n)$ where
${\displaystyle \alpha _{n}^{\prime }={\frac {(\rho _{1};q)_{n}(\rho _{2};q)_{n}(aq/\rho _{1}\rho _{2})^{n}\alpha _{n}}{(aq/\rho _{1};q)_{n}(aq/\rho _{2};q)_{n}}}}\alpha _{n}^{\prime }$
${\displaystyle \beta _{n}^{\prime }=\sum _{j\geq 0}{\frac {(\rho _{1};q)_{j}(\rho _{2};q)_{j}(aq/\rho _{1}\rho _{2};q)_{n-j}(aq/\rho _{1}\rho _{2})^{j}\beta _{j}}{(q;q)_{n-j}(aq/\rho _{1};q)_{n}(aq/\rho _{2};q)_{n}}}}$
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.
I am reading different papers and the book "q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra" by Andrews and I am currently asking myself, why are mathematicians over the moon about this lemma and why is it important for the field of number theory except for the Ramanujan Identity? Is it important/cool for the field of analytic number theory because you can express transcendental number in another way ? Let's say that's possible, why should I use these expressions above ?
Here are cool papers: