Level-13 theta function identities of Ramanujan and applications
Abstract
We give a new and simple proof of the three tredecic level theta function identities of Entry 8(i), Chapter 20 of Ramanujan’s Second Notebook by efficiently employing the quintuple product identity and Weierstrass three-term identity. Earlier this was proved by J. N. O’Brien (1965) and R. J. Evans (1990). O’Brien employed the techniques of Atkin and Swinnerton-Dyer (1953) and Atkin and Hussain (1958) who proved the corresponding identities related to levels 5, 7, and 11 by efficiently using the $n$th roots of unity. Evans proved them by employing the theory of modular forms. As an application of these identities, we deduce many interesting theta function identities analogous to those of Ramanujan’s identities, Lambert’s series identity of S. Cooper and D. Ye (2015), certain identities involving the parts of the 13-dissection of $(q;q)_\infty $, a few interesting partition identities, and several new non-trivial trigonometric identities.
