If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’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.
International Mathematical Forum
McLaughlin, J., & Zimmer, P. (2010). Some Applications of a Bailey-type Transformation. International Mathematical Forum, 5(61-64), 3007-3022. Retrieved from https://digitalcommons.wcupa.edu/math_facpub/76
Preprint version is available here.