Proofs of conjectures on Ramanujan-type series of level 3

Abstract

A Ramanujan-type series satisfies $$ \frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\left( \frac{1}{2} \right)_{n} \left( \frac{1}{s} \right)_{n} \left(1 - \frac{1}{s} \right)_{n} }{ \left( 1 \right)_{n}^{3} } z^{n} (a + b n), $$ where $s \in \{ 2, 3, 4, 6 \}$, and where $a$, $b$, and $z$ are real algebraic numbers. The level $3$ case whereby $s = 3$ has been considered as the most mysterious and the most challenging, out of all possible values for $s$, and this motivates the development of new techniques for constructing Ramanujan-type series of level $3$. Chan and Liaw introduced an alternating analogue of the Borwein brothers' identity for Ramanujan-type series of level $3$; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan-Liaw identity, via the use of Ramanujan's class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, using a limiting case of the Kummer-Goursat transformation, a new identity for evaluating $z$, $a$, and $b$ for Ramanujan-type series such that $s = 3$ and $z < 0$, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level $3$ with quartic values for $z < 0$, $a$, and $b$.