Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions

1. Problem Statement

The paper addresses two interconnected problems in mathematical analysis and number theory:

1. Fourier Analysis of Elliptic Functions: To compute the Fourier transforms of a specific class of meromorphic periodic functions formed by the ratio of Jacobi elliptic functions ($J$) and hyperbolic functions ($\sinh(\pi x)$ or $\cosh(\pi x)$). Specifically, the authors investigate functions of the form $f(x) = J(2K'x, k) / \sinh(\pi x)$ or $J(2K'x, k) / \cosh(\pi x)$, where $K$ and $K'$ are complete elliptic integrals of the first kind.
2. Analytic Properties of Double Series: To utilize these Fourier transforms to derive and analyze a new family of double series, termed Eisenstein-type series ($\zeta_{j,l}(s, \tau)$). The goal is to establish their analytic continuation, functional equations, and explicit values at integer points, analogous to the properties of the Riemann zeta function and Dirichlet L-functions.

2. Methodology

The authors employ a rigorous combination of complex analysis techniques:

• Contour Integration and Residue Calculus:

  • To compute the Fourier transforms, the authors define the transform operator with a slight imaginary shift ($\epsilon$) to avoid poles on the real axis.
  • They utilize rectangular contours in the complex plane and apply the Cauchy Residue Theorem.
  • A key technique involves exploiting the periodicity and transformation properties of Jacobi elliptic functions (e.g., shifting $x \to x \pm i/2$) to relate different transform pairs.
  • For functions with double poles (where zeros of the denominator coincide with poles of the numerator), they carefully compute residues involving derivatives.

• Mellin Transform Connection:

  • The authors apply the Mellin transform to the derived Fourier transform pairs.
  • They establish a correspondence between the Fourier cosine/sine transforms of the original functions and the Mellin transforms of the resulting series.
  • Theorems 1 and 2 are central to this methodology. They generalize the relationship between the Mellin transform of a function $g(x) = f(x)/\cosh(\pi x)$ (or $\sinh$) and its Fourier transform, proving that these transforms satisfy functional equations of the form $s \to 1-s$.

• Analytic Continuation and Modular Transformations:

  • The series are parameterized by $\tau$ (in the upper half-plane) rather than the elliptic modulus $k$, utilizing the relation $\tau = iK'/K$.
  • The authors use the asymptotic behavior of the Hurwitz zeta function and its alternating version to prove uniform convergence and analytic continuation of the double series.

3. Key Contributions

A. Fourier Transform Pairs (Tables 1 & 2)

The paper derives 24 explicit Fourier transform pairs involving Jacobi elliptic functions.

• Table 1 (Identities 1–18): These involve functions with simple poles. A significant finding is that for these cases, the Fourier transform of a function in the class remains within the same class (a ratio of a Jacobi function and a hyperbolic function), often with a change in the modulus from $k$ to $k'$ (the complementary modulus).
• Table 2 (Identities 19–24): These involve functions with double poles. The resulting transforms are more complex, involving integrals of squared elliptic functions (e.g., $\int dn^2(u) du$) and linear terms in $y$.
• Identity 25: A specific Fourier transform pair related to the logarithmic derivative of the Jacobi theta function $\theta_4$, which is crucial for the $\zeta_{0,0}$ series.

B. Eisenstein-Type Series ($\zeta_{j,l}(s, \tau)$)

The authors define sixteen new double series $\zeta_{j,l}(s, \tau)$ (indexed by $j, l \in \{0,1,2,3\}$) based on the Mellin transforms of the functions in the Fourier pairs.

• Definition:
  $$\zeta_{j,l}(s, \tau) := \sum_{n \in \mathbb{Z}}' \sum_{m \ge 0} \frac{(2a-1)^m (1-2b)^n}{[m + \frac{1}{2} + (n + \frac{d}{2})\tau]^s \sin(\pi(\frac{c}{2} + (n + \frac{d}{2})\tau))}$$
  where $a,b,c,d \in \{0,1\}$ determine the indices $j, l$.

C. Analytic Results (Propositions 1–10)

For the sixteen completed series $\Lambda_{j,l}(s, \tau)$ (defined by multiplying $\zeta_{j,l}$ by Gamma factors and powers of $\pi$), the authors prove:

1. Analytic Continuation: Each $\Lambda_{j,l}(s, \tau)$ extends to a meromorphic function on the complex plane $\mathbb{C}$. Most are entire; those with poles have a simple pole at $s=1$.
2. Functional Equations: They establish a functional equation connecting $\Lambda_{j,l}(s, \tau)$ with $\Lambda_{l,j}(1-s, -1/\tau)$. This reflects the modular symmetry of the underlying elliptic functions.
   $$\Lambda_{j,l}(s, \tau) + \frac{i}{\tau} \Lambda_{l,j}(1-s, -1/\tau) = \text{RHS involving } \Lambda(s) \text{ or } \Lambda(s, \chi_4)$$
3. Explicit Values: The authors provide closed-form expressions for $\Lambda_{j,l}(n, \tau)$ when $n$ is a positive even or odd integer. These values are expressed in terms of the complete elliptic integrals $K, K', E$ and their derivatives.

4. Results Summary

• 16 New Series: The paper introduces and fully characterizes 16 distinct Eisenstein-type series.
• Symmetry: The series exhibit a duality under the transformation $(s, \tau) \to (1-s, -1/\tau)$, generalizing the functional equation of the Riemann zeta function.
• Special Cases:
  • $\Lambda_{0,0}(s, \tau)$ is unique as it requires the logarithmic derivative of the theta function (Identity 25) for its derivation.
  • The values at integers $s=1, 2, 3, 4$ are computed explicitly, linking them to elliptic integrals and Euler numbers.
• Unresolved Cases: The authors note that 12 of the 24 Fourier pairs (those with poles on the real line) were not used to derive the 16 series. They propose that deforming the integration contour could extend their results to a new family of series (Eq. 34).

5. Significance

• Unification: The work bridges the gap between the theory of Jacobi elliptic functions, Fourier analysis, and the theory of Eisenstein series. It demonstrates how the periodicity of elliptic functions generates modular properties in double series.
• Generalization: It extends the classical results of Riemann (zeta) and Dirichlet (beta) functions to a broader class of functions involving elliptic moduli.
• Computational Utility: The explicit formulas for values at integers provide new tools for calculating specific constants related to elliptic integrals and theta functions.
• Future Directions: The paper identifies a path for extending these results to series involving functions with real-axis poles, suggesting a rich area for future research in analytic number theory.

In conclusion, this paper provides a comprehensive framework for understanding the analytic properties of a new class of double series by leveraging the Fourier transform properties of Jacobi elliptic functions, successfully establishing their functional equations and special values.