Ramanujan-type partial theta identities and conjugate Bailey pairs
In his lost notebook, Ramanujan recorded a number of partial theta identities such as those appearing in [\textit{G. E. Andrews} and \textit{B. C. Berndt}, Ramanujan's lost notebook. Part II. New York, NY: Springer (2009; Zbl 1180.11001), Entry 6.6.1]; among them, we have \[ \sum_{n \geq 0} \dfrac{(q^{n+1})_{n} q^{n}} {(-a)_{n+1}(-q/a)_{n}} = \sum_{n \geq 0} (-a)^{n} q^{n^{2}+n} - \dfrac{a}{(-a,-q/a)_{\infty}} \sum_{n \geq 0} (-1)^{n} a^{3n} q^{3n^2+2n}(1+aq^{2n+1}). \] Here we employ the usual basic hypergeometric series notation, namely \[ (a)_{n} := (a;q)_{n} = (1-a)(1-aq) \cdots (1-aq^{n-1}), \quad (a_{1},a_{2},\ldots, a_{j})_{n} : = (a_{1};q)_{n}(a_{2};q)_{n} \cdots (a_{j};q)_{n}, \] and \[ (a)_{\infty} = \lim_{n \to \infty} (a)_{n} \] when \(|q| < 1\). Ramanujan's partial theta identities were first proven by \textit{G. E. Andrews} [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001)]. Subsequently, \textit{S. O. Warnaar} [Proc. Lond. Math. Soc. (3) 87, No. 2, 363--395 (2003; Zbl 1089.05009)] placed the identities in the context of Bailey pairs and demonstrated how to generate further examples. A Bailey pair relative to \((a,\, q)\) is a pair of sequences \((\alpha_{n},\, \beta_{n})_{n \geq 0}\) satisfying \[ \beta_{n} = \sum_{r=0}^{n} \dfrac{\alpha_{r}}{(aq)_{n+r}(q)_{n-r}}. \] A conjugate Bailey pair relative to \((a,\, q)\) is a pair of sequences \((\delta_{n},\, \gamma_{n})_{n \geq 0}\) satisfying \[ \gamma_{n} = \sum_{r=n}^{\infty} \dfrac{\delta_{r}}{(aq)_{r+n}(q)_{r-n}}. \] The author demonstrates how Ramanujan-type partial theta identities give rise, through their residual identities, to conjugate Bailey pairs. To illustrate his ideas, the author proves six partial theta identities of Ramanujan and five identities stated by Warnaar; the author additionally derives six further identities using Warnaar's method. It is known [\textit{W. N. Bailey}, Proc. Lond. Math. Soc. (2) 50, 1--10 (1948; Zbl 0031.39203)] that a Bailey pair and a conjugate Bailey pair satisfy by a simple interchange of sums \[ \sum_{n \geq 0} \beta_{n} \delta_{n} = \sum_{n \geq 0} \alpha_{n} \gamma_{n}. \] The author records his results in a form corresponding to this relation. A typical result is Identity 1 of Theorem 1.1: \(\text{\textit{If}}\) \((\alpha_{n}, \, \beta_{n})\) \(\text{\textit{is}}\) \(\text{\textit{a}}\) \(\text{\textit{Bailey}}\) \(\text{\textit{pair}}\) \(\text{\textit{relative}}\) \(\text{\textit{to}}\) \((a, q)\), \(\text{\textit{then}}\) \[ \sum_{n \geq 0} q^{n} \beta_{n} = \dfrac{1}{(aq,q)_{\infty}} \sum_{r, n \geq 0} (-a)^n q^{\frac{n^2+n}{2} + (2n+1)r} \alpha_r. \] To prove his results, the author typically starts with a Ramanujan-type partial theta identity, computes the corresponding residual identity, and then deduces a statement about Bailey pairs. The author notes that some \(q\)-mixed mock modular forms \(\sum_{i=1}^{l} f_{i}g_{i}\), where \(f_{i}\) is a modular form and \(g_{i}\) is a mock modular form, may be derived by these methods, and he presents one such example to illustrate. A source of inspiration for the author's identities is cited as \textit{G. E. Andrews} and \textit{S. O. Warnaar} [Ramanujan J. 14, No. 1, 173--188 (2007; Zbl 1116.33017)], in which the partial theta identities \[ \sum_{n \geq 0} \dfrac{(a^2 q^2; q^4)_{n}(a^2q^2;q^2)_{n} q^n}{(a^2q^2,-aq^3,q^2;q^2)_{n}} = \dfrac{(-q)_{\infty}}{(-aq)_{\infty}} \sum_{n \geq 0} a^n q^{n^{2}} \tag{1} \] and \[ \sum_{n \geq 0} \dfrac{(aq)_{2n} q^{n}}{(a^2q^2, q^2; q^2)_{n}} = \dfrac{(-q)_{\infty}}{(-aq)_{\infty}} \sum_{n \geq 0} a^n q^{\frac{n^2+n}{2}} \tag{2} \] are used to deduce that if \((\alpha_{n}, \beta_{n})\) is a Bailey pair relative to \((a^2, q^2)\), then \[ \sum_{n \geq 0} \dfrac{(a^2q^2; q^2)_{2n} q^n}{(-aq)_{2n+1}} \beta_{n} = \dfrac{(-q)_{\infty}}{(-aq)_{\infty}} \sum_{r,n \geq 0} a^n q^{n^2 + n + 2rn + r} \alpha_{r} \] and \[ \sum_{n \geq 0} (aq)_{2n} q^n \beta_{n} = \dfrac{(-q)_{\infty}}{(-aq)_{\infty}} \sum_{r,n \geq 0} a^n q^{\frac{n^2+n}{2} + 2rn + r} \alpha_{r}. \] The investigation of whether there is a systematic way to produce ``good'' partial theta identities such as (1) and (2) led the author to the residual identities of Ramanujan-type partial theta identities.
scientific article; zbMATH DE number 6120461
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