Some Applications of a Bailey-type Transformation

Abstract: 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.