Some Applications of a Bailey-type Transformation

TL;DR

This paper explores a specific Bailey-type transformation that simplifies to a sum relation, leading to new hypergeometric identities, connections to classical problems, and novel series representations.

Contribution

It introduces new transformation formulas and identities derived from a Bailey pair with a special parameter choice, expanding the theory of basic hypergeometric series.

Findings

Derivation of several hypergeometric summation formulas

Connections established with the Prouhet-Tarry-Escott problem

New identities of Rogers-Ramanujan-Slater type and series representations

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.