Multiple integral representations of the Catalan's constant

Technical Summary: Multiple Integral Representations of Catalan's Constant

Problem Statement
Catalan's constant ($G \approx 0.915965$), defined by the alternating series $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}$, is a fundamental mathematical constant with numerous known integral and series representations. While existing literature (e.g., [1], [7], [9]) compiles extensive lists of such formulas, this paper aims to derive novel integral representations for $G$. Specifically, the authors seek to establish new forms involving double integrals, single integrals, and general multiple integrals of dimension $r \geq 2$, utilizing properties of cumulative distribution functions (CDFs) and the Lerch transcendent function.

Methodology
The paper employs a constructive analytical approach based on integration by parts and series summation. The core methodology relies on the following steps:

1. Foundation with CDFs: The authors define a class of functions $G_i: \mathbb{R} \to [0, 1]$ that are right-continuous, monotone non-decreasing, satisfy boundary conditions ($\lim_{x\to-\infty} G_i(x)=0$, $\lim_{x\to\infty} G_i(x)=1$), and possess the symmetry property $G_i(x) = 1 - G_i(-x)$. These conditions imply that the derivatives of these functions are even, a property crucial for eliminating odd terms during integration.
2. Derivation of Double Integrals: By integrating the product of powers of variables against these functions and applying integration by parts, the authors derive a general double integral identity. Summing the resulting series with alternating signs leads to the definition of $G$.
3. Extension to Single Integrals: A specific auxiliary function is introduced to reduce the double integral formulation into a single integral representation.
4. Generalization to Multiple Integrals: The approach is generalized to $r$ dimensions. The authors utilize the Lerch transcendent function, $\Phi(z, s, a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s}$, to express the sum of the series arising from $r$-dimensional integration.

Key Contributions and Results

• Theorem 1 (Double Integral Representation): The paper establishes that for any $a > 0$ and functions $G_1, G_2$ satisfying the stated conditions:
  $$G = \int_{-a}^{a} \int_{-1/a}^{1/a} \frac{G_1(x_1)G_2(x_2)}{1 + x_1^2 x_2^2} \, dx_1 dx_2$$
  This result generalizes known formulas, such as the classic $\int_0^1 \int_0^1 \frac{dx dy}{1+x^2y^2} = G$. The authors provide six illustrative examples using different CDFs, including the Rademacher distribution, the hyperbolic secant distribution, the standard normal distribution (via the error function), and the Cauchy distribution.

• Theorem 2 (Single Integral Representation): As a consequence of the double integral result, the authors derive a novel single integral representation:
  $$G = \int_{-1}^{1} G(x) \frac{\tan^{-1} x}{x} \, dx$$
  where $G(x)$ satisfies the conditions of Theorem 1. This yields specific new formulas, such as those involving the Cauchy CDF and the arcsin distribution.

• Theorem 3 (Multiple Integral Representation): The paper presents a general formulation for dimensions $r \geq 1$:
  $$G = 2^{r-2} \int_{-a_1}^{a_1} \cdots \int_{-a_r}^{a_r} \Phi\left(-\prod_{i=1}^r x_i^2, 2-r, \frac{1}{2}\right) \prod_{i=1}^r G_i(x_i) \, d\mathbf{x}$$
  where $a_r = (\prod_{i=1}^{r-1} a_i)^{-1}$. For $r=2$, this reduces to Theorem 1; for $r=1$, it reduces to Theorem 2.

• Explicit High-Dimensional Examples: The authors provide explicit integrands for dimensions $r=3$ through $r=10$ using the Rademacher distribution (where $G_i(x) = 1/2$ for $x \in [-1, 1]$).

  • For $r=3$, the integrand involves the rational function $\frac{1 - x(3)}{(1 + x(3))^2}$, where $x(3) = (x_1 x_2 x_3)^2$.
  • For higher dimensions ($r=4, 5, 6, 10$), the integrands are presented as polynomial rational functions involving coefficients derived from the expansion of the Lerch function. For instance, the $r=10$ case involves a polynomial with coefficients such as $a_1 = -6552$ and $a_2 = 331612$.

Significance and Claims
The paper claims to provide "several novel integral representations" of Catalan's constant. The significance lies in the unification of these representations under a single theoretical framework involving cumulative distribution functions and the Lerch transcendent. The authors demonstrate that by selecting different valid CDFs, one can generate a wide variety of integral formulas for $G$, ranging from simple double integrals to complex high-dimensional integrals with polynomial rational kernels. The work extends the known catalog of formulas for $G$ (referencing compilations like [1] and [7]) by introducing a systematic method to generate them in arbitrary dimensions. The paper does not propose new applications for these constants but focuses on the mathematical derivation and the structural relationships between different integral forms.