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

TL;DR

The paper addresses how Ramanujan's continued fractions of orders five, ten, and twenty interrelate with theta-function and Eisenstein-series structures. It develops a unified method based on the product expansion of the Jacobi theta function $\theta_1$ and introduces the auxiliary products $\Omega_k(q)$ to connect $T_1(q)$, $T_2(q)$ with Rogers–Ramanujan and order-ten fractions, yielding a network of identities $O1$–$O9$. It further derives Eisenstein-series identities of level twenty by exploiting $_1\psi_1$ summation and theta-derivative techniques, tying level-20 modular forms to the continued fractions and theta functions. These results extend earlier work on related fractions (orders 6, 12, 16) and provide a systematic framework for generating similar identities for other orders in the realm of $q$-series and modular forms.

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 $θ_1$, 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 $_1 ψ_1$ 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.

Theorems & Definitions (13)

• Lemma 2.1: Adiga_1985
• Theorem 3.1
• proof : Proof of \ref{['O1-O9']}
• proof : Proof of \ref{['1O1+9O9']}
• proof : Proof of \ref{['O99-O11']}
• Theorem 4.1
• proof
• Lemma 5.1
• proof
• Theorem 5.1