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Positive rational series for reciprocal powers of Catalan's constant and Dirichlet beta values

This paper introduces two families of positive rational series, derived via Euler and alternative transformations evaluated at $1/2$ and $1/3$ respectively, to efficiently compute reciprocal powers of Catalan's constant and other Dirichlet beta values, complete with explicit coefficients, convergence rates, and remainder estimates.

Original authors: Narendra Bhandari

Published 2026-08-04

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Narendra Bhandari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about numbers. In the vast library of mathematics, there are certain special numbers that act like the "DNA" of shapes and patterns. One of these is a number called Catalan's constant, often just called $G$ . It shows up when you add up a specific list of fractions involving odd numbers, but with a twist: you add one, subtract the next, add the next, and so on. It's like a seesaw that never quite settles. Mathematicians have known about this number for a long time, but they still don't know its true nature. Is it a simple fraction? Is it a weird, never-ending decimal that can't be written as a fraction? We don't know yet. It's one of those stubborn riddles that keeps mathematicians up at night.

To crack these riddles, scientists often try to write these mysterious numbers in different ways. Think of it like trying to describe a complex flavor of ice cream. You could say it's "sweet and cold," or you could list the exact grams of sugar, milk, and vanilla. In math, we have "series," which are just long lists of numbers added together. The problem is, the usual lists for Catalan's constant are messy. They jump back and forth between positive and negative numbers, and they get smaller very slowly, like a snail walking toward a finish line. This makes it hard to calculate the number precisely or to understand its secrets. So, the goal is to find a "cleaner" recipe: a list where every single number is positive, where the numbers are simple fractions, and where the list adds up to the answer incredibly fast.

This paper by Narendra Bhandari is like a master chef who has just invented two brand-new, super-efficient recipes for calculating the "reciprocal powers" of these special numbers. "Reciprocal" just means flipping the number upside down (like turning 2 into 1/2), and "powers" means multiplying it by itself a few times. The author isn't trying to solve the mystery of whether Catalan's constant is a fraction or not; instead, he is building better tools to measure it.

The paper presents two main tricks, or "transformations," to turn the messy, slow, jumping lists into fast, smooth, all-positive lists. The first trick is a classic method called the Euler transformation. Imagine you have a bumpy road (the old list) and you want to drive on a smooth highway. This method takes the bumpy road and paves it over, creating a new list where every term is a positive number. This new list adds up to the answer when you plug in the number 1/2. It works for almost any version of the problem, but it's a bit like driving a sedan: reliable, but not the fastest.

The second trick is the paper's real showstopper. It's a "super-charged" version of the first trick, designed for specific cases where the numbers are a bit more complex. This method creates a list that adds up even faster, using the number 1/3 as its key. The author proves that for these specific cases, every single number in the new list is not only positive but also a simple fraction (a rational number). This is huge because it means you can calculate these values using only basic arithmetic, without needing messy decimals.

The paper doesn't just stop at making the lists; it also gives a "recipe book" for how to generate the numbers in the lists. It provides a simple rule (a recurrence) that tells you how to get the next number in the list just by looking at the previous ones. It's like having a machine that spits out the next ingredient automatically. The author also proves that these new lists are incredibly accurate, giving a mathematical guarantee on how close the answer gets as you add more terms. In fact, the new lists converge (reach the answer) much faster than the old ones, shrinking the error by a factor of 2/3 with every single step.

Finally, the author shows how these new recipes can be used to calculate not just Catalan's constant, but also other famous numbers related to it, like powers of $\pi$ (the ratio of a circle's circumference to its diameter). By flipping the script and looking at the "upside-down" versions of these numbers, the paper provides a whole new family of positive, rational series. It's a toolkit for mathematicians who need to calculate these constants with extreme precision, offering a cleaner, faster, and more reliable way to explore the hidden patterns of numbers. While it doesn't solve the ultimate mystery of Catalan's constant's nature, it gives us a much sharper lens to look at it.

Technical Summary: Positive Rational Series for Reciprocal Powers of Catalan's Constant and Dirichlet Beta Values

Problem Statement
The paper addresses the construction of series representations for reciprocal powers of the Dirichlet beta function, $\beta(s)^{-r}$ , where $r \ge 1$ is an integer. Specifically, it targets Catalan's constant $G = \beta(2)$ and values of $\beta(s)$ at odd integers (which relate to powers of $\pi$ ). While the Dirichlet beta function is defined by an alternating series, and its reciprocal powers can be expressed via Dirichlet series involving the Möbius function, these classical representations suffer from two limitations: they contain terms with non-fixed sign patterns and they converge algebraically rather than geometrically. The objective is to develop ordinary power series with positive rational coefficients that exhibit geometric convergence, thereby providing efficient computational tools and new structural insights into these constants.

Methodology
The author employs a transformation-based approach rooted in the integral representation of the beta series. The core methodology involves three steps:

Integral Representation: The paper utilizes the function $F_s(t) = \sum_{k=0}^\infty (-1)^k t^k / (2k+1)^s$ , which admits an integral representation involving the Gamma function:
$F_s(t) = \frac{1}{\Gamma(s)} \int_0^1 \frac{(-\log x)^{s-1}}{1+tx^2} dx.$
This representation is valid for $\text{Re}(t) > -1$ .
Fractional-Linear Substitutions: Two distinct substitutions are applied to the variable $t$ to map the unit disk to the domain of analyticity:
- Ordinary Euler Transformation: The substitution $t = w/(1-w)$ maps $w=1/2$ to $t=1$ . This yields a series valid for all real $s > 0$ .
- Accelerated Transformation: The substitution $t = 2u/(1-u)$ maps $u=1/3$ to $t=1$ . This yields a series valid for $s \ge 2$ .
Reciprocal Power Construction: The transformed functions take the form $1 - D(z)$ , where $D(z)$ is a power series with positive coefficients. The reciprocal powers $(1 - D(z))^{-r}$ are then expanded using the generalized binomial theorem. The coefficients of these reciprocal series are derived via:
- A finite-sum formula involving binomial coefficients and alternating sums.
- A positive recurrence relation (Theorem 4.2) that allows efficient generation of coefficients.
- A composition formula involving partial exponential Bell polynomials (Theorem 4.3).

Key Contributions and Results

Two Families of Positive Series: The paper establishes two families of series for $1/\beta(s)^r$ :
- A series evaluated at $1/2$ (convergence rate $1/2$ ) valid for $s > 0$ .
- A series evaluated at $1/3$ (convergence rate $1/3$ ) valid for $s \ge 2$ .
  Both families possess strictly positive coefficients. When $s$ is an integer, all coefficients are rational.
Explicit Formulas for Catalan's Constant ( $G$ ): For $s=2$ , the accelerated transformation provides a rapidly converging series for $1/G^r$ . The paper provides explicit formulas for the base coefficients $p_{2,n}$ and the reciprocal coefficients $c_{2,r,n}$ .
- Example: $1/G = 1 + \sum_{n=1}^\infty \frac{c_{2,1,n}}{3^n}$ , where the first few terms are $2/27, 226/18225, \dots$ .
- The paper proves that every term after the initial 1 is a positive rational number, ensuring that partial sums are strict lower bounds.
Reciprocal Powers of $\pi$ : By combining the accelerated transformation with the known relationship between odd beta values and powers of $\pi$ (involving Euler numbers), the paper derives positive rational series for $\pi^{-(2m+1)r}$ .
- For the case $\beta(1) = \pi/4$ , the ordinary Euler transformation (evaluated at $1/2$ ) is used, as the accelerated transformation fails to produce positive coefficients at $s=1$ .
- This yields series such as $4^r/\pi^r = 1 + \sum_{n=1}^\infty \frac{c_{1,r,n}}{2^n}$ .
Convergence and Error Analysis:
- The exact root-convergence rates are determined to be $1/2$ and $1/3$ for the two families, respectively.
- Explicit remainder estimates are provided (Theorem 4.7), showing geometric decay of the error. For instance, the error for the accelerated series is bounded by terms proportional to $(2/3)^{N+1}$ .
- Numerical comparisons confirm that the accelerated series converges significantly faster than the ordinary Euler series (e.g., at $N=80$ , the error for $1/G$ is $\approx 10^{-41}$ for the accelerated series versus $\approx 10^{-27}$ for the ordinary series).
Coefficient Properties: The paper details the structure of the coefficients, showing they can be generated via a simple recurrence $c_n^{(r)} = \frac{1}{n} \sum_{k=1}^n [n + (r-1)k] d_k c_{n-k}^{(r)}$ . It also provides closed-form expressions involving Bell polynomials and generalized harmonic numbers for integer $s$ .

Significance and Claims
The paper claims to provide a systematic method for generating positive rational series for reciprocal powers of Catalan's constant and related beta values. Unlike previous approaches that rely on modular forms or hypergeometric identities (Ramanujan-type series), this method is "Euler-type," relying on classical series transformations and integral representations.

The significance lies in:

Computational Efficiency: The geometric convergence rates ( $1/2$ and $1/3$ ) and the positivity of terms allow for efficient high-precision calculation of these constants and their reciprocals.
Structural Clarity: The derivation of finite-sum formulas and positive recurrences offers a clear algebraic structure for these coefficients, which are otherwise difficult to characterize in the context of reciprocal powers.
Rationality: The explicit demonstration that coefficients are rational when $s$ is an integer provides a new class of rational approximations for these constants.

The author explicitly notes that the work does not aim to resolve the irrationality of $G$ or other open arithmetic problems but rather focuses on the analytic and combinatorial construction of these specific series representations. The results are presented as self-contained analytical tools with immediate applications in numerical evaluation.

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