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A Regularised Wallis Hierarchy

This paper introduces a hierarchy of regularised Wallis products by raising reciprocal Wallis factors to polynomial weights and cancelling non-summable logarithmic terms with minimal exponential counterterms, yielding convergent products expressible via zeta-function tails, harmonic numbers, and multiple sine functions.

Original authors: S. R. Holcombe

Published 2026-06-24

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: S. R. Holcombe

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 have a classic, perfectly balanced recipe for a mathematical cake known as Wallis' Product. For centuries, mathematicians have known how to bake this cake to get the number $\pi$ . The recipe involves multiplying a long list of simple fractions together.

This paper, written by S. R. Holcombe, asks a "what if" question: What happens if we change the recipe?

Instead of just multiplying the fractions once, what if we raise them to higher and higher powers (like squaring them, cubing them, etc.)?

The Problem: The Cake Explodes

If you simply take those fractions and raise them to a high power (like $n^2$ or $n^3$ ), the recipe breaks. The numbers get so big or so small that the total sum doesn't settle on a specific answer; it "explodes" into infinity or zero. It's like trying to bake a cake where the ingredients keep doubling in size every time you stir—the batter never stops growing.

The Solution: The "Regularized" Fix

The author introduces a new method called a Regularised Wallis Hierarchy. Think of this as a mathematical "tuning fork" or a noise-canceling headphone for the recipe.

The Noise: When the recipe explodes, it's because of specific "loud" terms in the math that grow too fast.
The Counter-Recipe: The author creates a tiny, precise "counter-ingredient" (an exponential factor) for each step. This counter-ingredient is designed to perfectly cancel out the "loud" noise, but only the noise. It leaves the "music" (the actual meaningful part of the product) untouched.
The Result: Once the noise is cancelled, the infinite list of numbers finally settles down into a clean, finite, and beautiful answer.

The Two Branches of the Family

The author organizes these new recipes into a family tree with two distinct branches, based on whether the power used is an even number (2, 4, 6...) or an odd number (1, 3, 5...).

The Even Branch (The $\pi$ Family):
When the power is even, the final answer is a "closed form" recipe. It's a neat, tidy expression involving the number $\pi$ , harmonic numbers (a specific type of sum), and odd "zeta values" (famous mathematical constants).
- Example: For the power of 2, the recipe yields $\frac{\pi}{e^{3/2}}$ . It's a clean, elegant result.
The Odd Branch (The $\gamma$ Family):
When the power is odd, the answer is a bit more complex. It involves the Euler-Mascheroni constant ( $\gamma$ , a famous number in math), logarithms, and derivatives of the zeta function. It's like a recipe that requires a few more exotic spices, but it still results in a precise, finite number.
- Example: For the power of 1, the recipe yields $\frac{e^\gamma}{2}$ .

The "Hyperbolic" Cousin

The author also experiments with a "twin" recipe. Instead of using the fraction $(1 - \frac{1}{n^2})$ , they use its "hyperbolic cousin" $(1 + \frac{1}{n^2})$ .

Imagine the original recipe uses a "minus" sign, and this new one uses a "plus" sign.
The author shows that you can mix and match these two recipes (using both plus and minus versions) and still use the same "noise-canceling" technique to get clean answers.

The Connection to "Multiple Sine" Functions

Finally, the paper connects these new recipes to a grander mathematical structure called Kurokawa's Multiple Sine Functions.

Think of the original Wallis product as a single note.
The author shows that these new, complex recipes are actually just different ways of looking at these "Multiple Sine" instruments.
The "Even" branch of the new recipes corresponds to "Odd" multiple sine functions, and the "Odd" branch corresponds to "Even" multiple sine functions. It's like discovering that two different musical scales are actually playing the same song, just in different keys.

Summary

In simple terms, this paper takes a famous, simple math formula, turns the "volume" up by raising it to higher powers, and then invents a clever "mute button" (the counter-term) to stop it from blowing up. This reveals a whole new hierarchy of beautiful, finite numbers that were previously hidden by the mathematical noise. It connects these new numbers to famous constants like $\pi$ and $\gamma$ , and links them to a broader family of mathematical functions known as multiple sines.

Technical Summary: A Regularised Wallis Hierarchy

Problem Statement
The paper addresses the convergence properties of infinite products derived from the reciprocal Wallis factor, $1 - 1/n^2$ . While the classical Wallis product and its telescoping reciprocal counterpart converge with constant exponents, raising the reciprocal factor to a polynomial weight $n^m$ (where $m = 0, 1, 2, \dots$ ) destroys ordinary convergence for $m \geq 1$ . The logarithm of such factors generates divergent series terms. The central problem is to define a consistent regularisation scheme that removes these divergences while preserving the structural relationship to the original Wallis factor, thereby creating a hierarchy of convergent products $P_m$ .

Methodology
The author introduces a hierarchy of regularised products $P_m$ by applying a minimal subtraction rule to the logarithmic expansion of the terms.

Logarithmic Expansion: The term $\log(1 - 1/n^2)$ is expanded as $-\sum_{j=1}^\infty \frac{1}{j n^{2j}}$ . Multiplying by the weight $n^m$ yields $-\sum_{j=1}^\infty \frac{n^{m-2j}}{j}$ .
Identification of Divergence: The terms $n^{m-2j}$ are non-summable when $m - 2j \geq -1$ . The cutoff index is defined as $J_m = \lfloor \frac{m+1}{2} \rfloor$ .
Regularisation: A minimal exponential counterterm is introduced to cancel precisely the non-summable terms in the expansion. The regularised product is defined as:
$P_m = \prod_{n=2}^\infty \exp\left( \sum_{j=1}^{J_m} \frac{n^{m-2j}}{j} \right) \left( 1 - \frac{1}{n^2} \right)^{n^m}$
Zeta-Tail Formulation: After cancellation, the logarithm of the product reduces to an absolutely convergent double series, which is rearranged to express $\log P_m$ explicitly in terms of tails of the Riemann zeta function, $\zeta(s)$ .
Branch Separation: The hierarchy is analyzed by separating even ( $m=2a$ ) and odd ( $m=2a+1$ ) indices, leading to distinct closed-form evaluations involving different classes of special functions.

Key Contributions and Results

Explicit Zeta-Tail Representation: The primary result is the derivation of $\log P_m$ as a sum involving $\zeta(k) - 1$ .
- For even $m=2a$ : $\log P_{2a} = -\sum_{r=1}^\infty \frac{\zeta(2r)-1}{r+a}$ .
- For odd $m=2a+1$ : $\log P_{2a+1} = -\sum_{r=1}^\infty \frac{\zeta(2r+1)-1}{r+a+1}$ .
Closed-Form Evaluation of the Even Branch:
The even branch is evaluated in terms of elementary constants, harmonic numbers ( $H_n$ ), and odd zeta values. The result is:
$P_{2a} = \pi \exp\left[ -H_{a-1/2a} + 2a \sum_{\ell=1}^{a-1} \frac{(-1)^{\ell+1}(2a-1)!}{(2a-2\ell)!} \frac{\zeta(2\ell+1)}{(2\pi)^{2\ell}} \right]$
(Note: The paper expresses the harmonic term as $-H_a - 1/2a$ in the final formula).
Closed-Form Evaluation of the Odd Branch:
The odd branch reduces to finite logarithmic gamma moments. The evaluation involves the Euler-Mascheroni constant $\gamma$ , logarithms, odd zeta values, and derivatives of the zeta function at positive even integers ( $\zeta'(2k)$ ). The result is expressed via moments $M_k = \int_0^1 x^k [\psi(1+x) + \gamma] dx$ .
Hyperbolic Companion and Two-Factor Extension:
The methodology is extended to include the hyperbolic companion factor $(1 + 1/n^2)$ . A two-parameter family $P_m(A, B)$ is defined, where $P_m = P_m(1, 0)$ and the pure hyperbolic case is $P_m(0, 1)$ . The logarithm of the general case is shown to be linear in $A$ and $B$ : $\log P_m(A, B) = A \log P_m + B \log P_m^+$ .
Connection to Kurokawa's Multiple Sine Functions:
The paper establishes a direct link between the regularised Wallis products and Kurokawa's multiple sine functions, $S_r(z)$ .
- The even branch ( $m=2a$ ) is derived from the derivative of the odd multiple sine function $S_{2a+1}(z)$ at $z=1$ . Specifically, $P_{2a}$ is related to $S'_{2a+1}(1)$ .
- The odd branch ( $m=2a+1$ ) is identified as the symmetric companion to the even multiple sine case, arising from the product of one-sided canonical factors $\Phi_m(x)\Phi_m(-x)$ .

Specific Examples
The paper provides explicit evaluations for the first few members of the hierarchy:

$P_0 = 1/2$ (Classical telescoping product).
$P_1 = e^\gamma / 2$ .
$P_2 = \pi / e^{3/2}$ (recovering a result previously identified by Allouche via triple sine functions).
$P_3$ and $P_4$ are given in terms of $\gamma$ , $\zeta(3)$ , and $\zeta'(2)$ .

Significance and Claims
The paper claims to place isolated Wallis-type product identities into a unified, indexed family. By applying a fixed minimal-subtraction rule to the same Wallis factor for every $m$ , the author constructs a "Regularised Wallis Hierarchy" where the logarithms are explicit zeta tails.
The significance lies in:

Unification: Connecting previously disparate results (such as the $m=2$ case found by Allouche) into a single framework.
Structural Insight: Revealing that the even and odd branches of the hierarchy correspond naturally to Kurokawa's odd and even multiple sine functions, respectively.
Explicit Formulas: Providing closed-form expressions for these products that involve standard mathematical constants and special values of the zeta function, extending the known landscape of regularised infinite products.

The author notes that the formulation is distinct from other generalisations (e.g., by Cai, Hu, Kim, or Farrell) because it strictly adheres to the minimal-subtraction rule on the specific reciprocal Wallis factor, yielding a hierarchy with explicit zeta-tail logarithms. The paper does not propose new applications or experimental directions but focuses on the analytical derivation and classification of these product identities.