Enhancing Theoretical Foundations in modular Forms: New Relations Between Eisenstein Series and Ramanujan’s k-functions

The sophisticated correlation between the Eisenstein series and theta functions was recorded by Ramanujan in his notebook. Inspired by this, in this article, we develop novel relations pertaining to Eisenstein series and Ramanujan’s k-functions that show the sum of Eisenstein series converges to Ramanujan’s k-functions. Additionally, we build new and appealing differential identities that connect Dedekind functions with k-functions using these connections. We conclude by demonstrating that the sum of the Class One infinite series converges to k-functions and assess a discrete convolution sum by utilizing the convergence criteria of the Eisenstein series.