On certain new Ramanujan type theta function identities

Abstract

In the scattered places of his notebook, Ramanujan recorded identities involving theta functions $f(-q^{\frac{1}{p}})$, $f(-q^p)$, and $f(-q^a,-q^b)$, with $a+b=p$, for $p=5,7,13,17$, etc. These identities have been studied by many mathematicians and their applications to the theory of partition, non-trivial trigonometric identities, Lambert's series identities associated with theta functions, and many more have also been explored. In this article, our aim is to establish analogous theta function identities for $p=25$ and 50. Further, we deduce certain relations among the parts of the 25-dissection of $f(-q)$ and the parts of the 50-dissection of $\phi \left(q\right)$ and $\psi \left(q\right)$. Finally, we deduce certain interesting non-trivial finite trigonometric identities and a few partition identities as applications of our main results.