\(q\)-hypergeometric double sums as mock theta functions (Q363336)

A Bailey pair relative to \(a\) is a pair of sequences \((\alpha_n, \beta_n)_{n \geq 0}\) satisfying NEWLINE\[NEWLINE\beta_n=\sum_{k=0}^n \frac{\alpha_k}{(q)_{n-k}(aq)_{n+k}}NEWLINE\]NEWLINE where \((a)_n=(a;q)_n=\prod_{k=1}^n(1-aq^{k-1})\).NEWLINENEWLINEThe authors use some techniques involving the Bailey lemma to construct two new Bailey pairs from two such pairs given by \textit{K. Bringmann} and \textit{B. Kane} [Trans. Am. Math. Soc. 363, No. 4, 2191--2209 (2011; Zbl 1228.11157)]. The new pairs give rise to four new genuine mock theta functions (as opposed to the pairs given by Bringmann and Kane that lead to mixed mock modular forms). To prove that the arising series are mock theta functions, the authors use recent results of \textit{D. R. Hickerson} and \textit{E. T. Mortenson} [Proc. Lond. Math. Soc. (3) 109, No. 2, 382--422 (2014; Zbl 1367.11047)] to relate the series to Appell-Lerch series. Finally, one of the new functions is expressed in terms of classical eight-order mock theta functions.