On Eisenstein series identities and new identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four

1. Problem Statement

The paper addresses the challenge of establishing new algebraic and analytic relationships between two specific types of continued fractions central to the theory of $q$-series and modular forms:

1. The Ramanujan-Göllnitz-Gordon (RGG) continued fraction, denoted as $H(q)$.
2. Ramanujan's continued fraction of order four, denoted as $I(q)$.

While previous literature has explored relations between $H(q)$ and its powers (e.g., $H(q^n)$) or relations involving other continued fractions (like the Rogers-Ramanujan or cubic fractions), there was a need for a systematic framework to derive direct identities connecting $H(q)$ and $I(q)$ using theta function evaluations. Additionally, the authors sought to derive new Eisenstein series identities associated with these fractions.

2. Methodology

The authors employ classical tools from the theory of $q$-series, theta functions, and modular forms. The methodology is structured as follows:

• Theta Function Product Expansions: The core technique involves utilizing the product expansion of Jacobi's theta function, $\theta_1(z|\tau)$. Specifically, the authors evaluate $\theta_1$ at special arguments $z = \frac{\pi}{8}, \frac{2\pi}{8}, \frac{3\pi}{8}$.
• Infinite Product Construction: By evaluating $\theta_1$ at these specific points, the authors define three infinite products, $\Gamma_1(q), \Gamma_2(q), \Gamma_3(q)$, involving coefficients $\beta_k$ derived from the fourth roots of unity (specifically $\beta_1 = -\sqrt{2}, \beta_2 = 0, \beta_3 = \sqrt{2}$).
• Algebraic Manipulation: The authors perform linear combinations of these $\Gamma$ functions (e.g., $\Gamma_1 - \Gamma_3$, $\beta_3\Gamma_3 - \beta_1\Gamma_1$). They utilize MAPLE computations to determine the coefficients of the resulting $q$-series expansions, which are then re-expressed using Ramanujan's general theta function $f(\gamma, \delta)$.
• Connection to Continued Fractions: The authors relate these manipulated theta products back to the definitions of $H(q)$ and $I(q)$ via the Dedekind eta function $\eta(\tau)$ and known product representations.
• Ramanujan's $1\psi_1$ Summation: In the second part of the paper, the authors utilize Ramanujan's $1\psi_1$ summation formula to derive identities involving Eisenstein series. They manipulate summations over odd integers and specific modular congruences to link series expansions to the continued fractions.

3. Key Contributions and Results

A. Identities Connecting $H(q)$ and $I(q)$

The primary contribution is the derivation of a family of six new identities (Theorem 3.1) that express combinations of the infinite products $\Gamma_k$ in terms of ratios involving $H(q)$ and $I(q)$.

• Theorem 3.1: Establishes relations such as:
  $$\Gamma_1^2(q^{1/2})\Gamma_3(q^{1/2}) - \Gamma_1(q^{1/2})\Gamma_3^2(q^{1/2}) = C(q) \cdot \frac{\sqrt[4]{I(q)}}{\sqrt{H(q)}}$$
  where $C(q)$ is a complex factor involving powers of $q$ and the Dedekind eta function $\eta(\tau)$.
• Significance: These identities provide explicit algebraic links between the RGG fraction and the order-four fraction, allowing for the evaluation of one in terms of the other and their associated modular forms.

B. Eisenstein Series Identities

The authors derive several new identities expressing Eisenstein series (infinite sums involving $q$) in terms of the continued fractions and theta functions.

• Theorem 4.1: Establishes an identity for a specific difference of series involving $q^m, q^{3m}, q^{5m}, q^{7m}$ over $1-q^{8m}$, equating it to a product of eta functions and the term $\left[\frac{1}{H(q^2)} + H(q^2)\right]$.
• Theorems 4.3 & 4.4: Extend these results to include Legendre symbols and different modular congruences (mod 4, mod 8), linking them to ratios of $\phi(q)$ and $\psi(q)$ (Ramanujan's theta functions).
  • Example: $\sum \frac{m(q^m - q^{3m} - q^{5m} + q^{7m})}{1-q^{8m}} = \text{Eta Product} \times \left[\frac{\phi(q)}{q^{1/2}\psi(q^4)}\right]$.

4. Significance and Impact

• Unification of Continued Fractions: The paper successfully bridges the gap between the Ramanujan-Göllnitz-Gordon fraction and the order-four fraction, two distinct objects in the theory of special functions. This expands the known landscape of modular equations and continued fraction relations.
• Systematic Framework: The method of evaluating $\theta_1$ at specific rational multiples of $\pi$ to generate relations between different orders of continued fractions offers a reproducible technique for future research.
• New Eisenstein Series: The derived identities provide new representations for Eisenstein series of level 8, 16, and 32, connecting them directly to combinatorial partition identities and continued fractions.
• Validation of Classical Tools: The work demonstrates the continued power of classical $q$-series tools (Jacobi triple product, Ramanujan's $1\psi_1$) in solving modern problems in analytic number theory.

5. Conclusion

Bhat and Srivatsa Kumar have successfully utilized the product expansions of Jacobi's theta function to derive a comprehensive set of new identities linking the Ramanujan-Göllnitz-Gordon continued fraction with Ramanujan's continued fraction of order four. Furthermore, by applying Ramanujan's $1\psi_1$ summation, they established novel Eisenstein series identities. These results enrich the theory of $q$-series and provide deeper insights into the modular properties of Ramanujan's continued fractions.