Some Applications of a Bailey-type Transformation

Abstract

If $k$ is set equal to $aq$ in the definition of a WP Bailey pair, \[ β_{n}(a,k) = \sum_{j=0}^{n} \frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}α_{j}(a,k), \] 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.

18 pages