Elementary proofs of two theta function identities of Ramanujan and corresponding Lambert series identities
Volume 152 / 2018
Colloquium Mathematicum 152 (2018), 15-22
MSC: Primary 11P84; Secondary 05A17.
DOI: 10.4064/cm7019-8-2017
Published online: 12 January 2018
Abstract
We first present a simple proof of two theta function identities of Ramanujan by employing certain relations satisfied by Ramanujan’s cubic continued fraction. Then we establish an elementary proof of the corresponding Lambert series identities. Finally, we give an alternative proof of the recent relation between the number of 3-cores of a nonnegative integer $n$ and the number of representations of $n$ in the form $x^2+3y^2$ with $x,y\in \mathbb {Z}$, established by Baruah and Nath.
