  If $k$ is set equal to $a q$ in the definition of a WP Bailey pair, \[
\beta_{n}(a,k) = \sum_{j=0}^{n}
\frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}\alpha_{j}(a,k), \] this
equation reduces to $\beta_{n}=\sum_{j=0}^{n}\alpha_{j}$. This seemingly
trivial relation connecting the $\alpha_n$'s with the $\beta_n$'s has some
interesting consequences, including several basic hypergeometric summation
formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities
of the Rogers-Ramanujan-Slater type, some new expressions for false theta
series as basic hypergeometric series, and new transformation formulae for
poly-basic hypergeometric series.
