On a new parameter involving Ramanujan's theta-functions

TL;DR

This paper introduces a new parameter based on Ramanujan's theta-functions, explores its modular relations with existing parameters, and uses it to derive explicit evaluations of ratios of theta functions.

Contribution

It defines a novel parameter $A'_{k,n}$ related to Ramanujan's theta-functions and establishes modular relations to evaluate theta function ratios explicitly.

Findings

Derived explicit values of $A'_{k,n}$ using modular relations.

Established theorems for evaluating ratios of theta functions involving $$,

Connected the new parameter to existing ones for broader applicability.

Abstract

We define a new parameter $A'_{k,n}$ involving Ramanujan's theta-functions for any positive real numbers $k$ and $n$ which is analogous to the parameter $A_{k,n}$ defined by Nipen Saikia \cite{NS1}. We establish some modular relation involving $A'_{k,n}$ and $A_{k,n}$ to find some explicit values of $A'_{k,n}$. We use these parameters to establish few general theorems for explicit evaluations of ratios of theta functions involving $\varphi(q)$.