Multiple integral representations of the Catalan's constant

Abstract: In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem that enables the representation of Catalan's constant in terms of a single integral. Finally, we provide a multiple integral representation of Catalan's constant in dimensions greater than or equal to two using the Lerch function. The results are accompanied by illustrative examples.

• The paper presents a unifying framework that generates multiple integral representations for Catalan’s constant using cdf parameterizations and the Lerch transcendent.
• The paper employs double, single, and r-fold integral formulations with explicit examples from distributions like uniform, hyperbolic secant, and Gaussian.
• The paper’s framework bridges analytic number theory and probability, offering new avenues for high-precision numerical evaluations and theoretical generalizations.

Multiple Integral Representations of Catalan's Constant

Introduction

The paper "Multiple integral representations of the Catalan's constant" (2605.08899) addresses the construction of a suite of new integral representations for Catalan's constant $G$. The constant, crucial in diverse analytic and number-theoretic contexts, is usually encountered as an alternating series or in several integral forms. The contributions in this work include general methods for expressing $G$ with double, single, and higher-dimensional integrals, demonstrating a unifying framework that links probability theory, special functions, and analytic number theory.

Double Integral Formulations

The initial major result is the derivation of a family of double integral representations for $G$ parameterized by pairs of cumulative distribution functions (cdfs), $G_1(x_1)$ and $G_2(x_2)$. Given two right-continuous monotone non-decreasing functions on $\mathbb{R}$, symmetric about the origin, the authors show:

$$G = \int_{-a}^a \int_{-1/a}^{1/a} \frac{G_1(x_1) G_2(x_2)}{1 + x_1^2 x_2^2} dx_1 dx_2$$

for any $a>0$, under mild symmetry and normalization constraints for $G_1$ and $G_2$.

Several explicit instantiations using distribution functions—including the Rademacher distribution, the hyperbolic secant, and the Gaussian error function—demonstrate the breadth of the construction. Notably, these examples recover celebrated identities already present in the literature and furnish new representations. For instance, letting $G$0 be the uniform distribution on $G$1 yields the classical

$G$2

and other choices of cdf provide connections to the hyperbolic secant law and the Gaussian.

Single Integral Reductions

Leveraging the symmetry and suitable choice of cdfs, the authors reduce some of the double integral representations to single integrals involving elementary and special functions. They prove a general theorem:

$G$3

where $G$4 is any function satisfying the earlier cdf constraints. This not only recovers the classical Ramanujan integral

$G$5

but also extends the catalogue to include forms involving the cdf of the Cauchy law, the normal law, the arcsin law, and more. Consequently, the framework naturally relates the analytic and probabilistic perspectives on $G$6, bridging these approaches via integral transforms of distribution functions.

High-dimensional Integral Formulations and Lerch Transcendent

Advancing beyond double and single integrals, the central contribution is the generalization to multiple integrals in arbitrary dimensions. The authors provide a systematic method for expressing $G$7 as an $G$8-fold integral involving products of arbitrary cdfs and rational functions, with normalization given in terms of the Lerch transcendent $G$9. For instance, the triple integral form reads:

$G$0

with precise parameterizations and normalization dictated by the Lerch function.

The approach is systematically extended, leading to compact representations of $G$1 in up to ten dimensions. In each case, the integrand involves polynomial or rational functions with explicit coefficients, supported either on hypercubes $G$2 or $G$3. The intricate structure of the denominator is connected to the expansion of the Lerch function and correlates with the combinatorial complexity of the multiple integral.

Theoretical and Practical Implications

The framework subsumes traditional representations and provides a mechanism to generate new integral identities. The methodology shows that the specific algebraic structure of the cdf determines the nature of the resulting integral, blending probabilistic and analytic perspectives. The connections to the Lerch transcendent explicitly link the evaluation of $G$4 to the analytic continuation and special values of polylogarithmic-type functions, underlining the deep interplay between special functions, probability, and number theory.

Practically, these multi-dimensional representations might facilitate new approaches to numerical evaluation or theoretical analysis of $G$5—for example, via probabilistic simulation or analytic summation techniques. The availability of multiple integral forms can be leveraged in high-precision computations, and the presence of parameters (such as $G$6 or the choice of cdfs) offers flexibility for optimizing convergence or cancellation properties.

Theoretically, the paper opens the possibility of generalizing this framework to other constants with polylogarithmic or Dirichlet–beta origins and suggests a categorical unification of integral representations for special numbers via probabilistic symmetries.

Conclusion

The paper presents a systematic and unifying approach to generating multiple, double, and single integral representations for Catalan's constant. By parameterizing integrals with cdfs and employing the Lerch transcendent for normalization, the authors establish a flexible framework that consolidates various analytic and probabilistic representations of $G$7. These results not only enrich the catalogue of explicit integral forms for $G$8 but also suggest deeper structural connections between analytic number theory, probability, and the theory of special functions, potentially informing future research on transcendental numbers and their representations.