On Ramanujan's Continued Fractions of Orders Five, Ten, and Twenty and Associated Eisenstein Series Identities

TL;DR

This paper establishes new identities linking Ramanujan's continued fractions of order twenty with Eisenstein series, theta functions, and other $q$-series, extending previous work on related orders and deepening the understanding of modular forms.

Contribution

It introduces novel relations connecting continued fractions of order twenty with Eisenstein series, theta functions, and other $q$-series, expanding the theory of modular forms and Ramanujan's identities.

Findings

Derived new relations between continued fractions of order twenty and order ten.

Established identities connecting Eisenstein series with theta functions of level twenty.

Extended earlier work on continued fractions of orders 6, 12, and 16.

Abstract

Eisenstein series play an important role in the theory of modular forms and have profound connections with $q$-series identities, partition theory, and special functions. Likewise, Ramanujan's mock theta functions, originally introduced in his last letter to Hardy, have inspired generations of mathematicians to work on $q$-series and modular forms. In this work, we establish several new identities connecting Ramanujan's continued fractions of order twenty. By employing product representation for Jacobi's theta function $\theta_1$, we derive a family of new relations connecting the continued fractions of order twenty with continued fractions of order ten and Rogers-Ramanujan continued fraction. Further, by expressing Eisenstein series in terms of Lambert series and utilizing certain mock theta functions and their logarithmic derivatives, we obtain beautiful relations between Eisenstein series and theta functions of level twenty. Using Ramanujan's $_1 \psi_1$ summation formula, we establish Eisenstein series identities associated with the continued fractions of order twenty. These results extend earlier work on continued fractions of order 6, 12, and 16 and contribute to theory of $q$-series.