[Paper Review] The regularized product of the Fibonacci numbers

Adrian R. Kitson|ArXiv.org|Aug 8, 2006

Advanced Mathematical Theories and Applications3 citations

TL;DR

This paper computes the regularized product of all Fibonacci numbers using zeta-function regularization, deriving an analytic expression for the infinite product $\Delta = \prod_{n=1}^{\infty} f_n$ via the derivative of the Fibonacci zeta function $\zeta_F(s)$. The key result is $\Delta = \frac{5^{1/4} \exp\left(-\ln^2(5)/(8\ln\phi)\right) c}{\phi^{1/12}}$, where $c$ is the Fibonacci factorial constant, yielding a numerical value of approximately 0.8992126807.

ABSTRACT

The regularized product of the Fibonacci numbers is evaluated.

Motivation & Objective

• To define and compute the regularized infinite product of all Fibonacci numbers, which diverges in the conventional sense.
• To extend zeta-function regularization techniques—previously applied to integers and primes—to the Fibonacci sequence.
• To resolve the divergence of the zeta derivative at $s=0$ by isolating and removing the principal part via asymptotic expansion.
• To express the regularized product in terms of known mathematical constants, including the Fibonacci factorial constant and the Jacobi theta function.
• To provide a numerically precise value for the regularized product of the Fibonacci sequence.

Proposed method

• Define the regularized product via $\Delta = \exp(-\zeta_F'(0))$, where $\zeta_F(s) = \sum_{n=1}^\infty f_n^{-s}$ is the Fibonacci zeta function.
• Use Binet's formula $f_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$ to approximate $f_n$ for regularization.
• Decompose $\zeta_F'(s)$ into a divergent leading term and a convergent remainder to handle the pole at $s=0$.
• Expand $\zeta_F'(s)$ in a Laurent series around $s=0$, isolating the finite part as $\zeta_F'(0)$.
• Simplify the constant term using the infinite product $\prod_{n=1}^\infty \left(1 - (-\phi)^{-2n}\right) = c$, the Fibonacci factorial constant.
• Express the final result in terms of the Jacobi theta function via $c = \left(\frac{1}{2}\vartheta_1'\left(0, -i/\phi\right)\right)^{1/3}$.

Experimental results

Research questions

• RQ1 What is the regularized value of the infinite product $\prod_{n=1}^\infty f_n$ when the standard product diverges?
• RQ2 How can zeta-function regularization be adapted to sequences like the Fibonacci numbers with exponential growth?
• RQ3 What is the finite part of $\zeta_F'(s)$ at $s=0$, after removing the pole?
• RQ4 Can the regularized product be expressed in terms of known special functions or constants?
• RQ5 What is the numerical value of the regularized product of all Fibonacci numbers?

Key findings

• The regularized product of all Fibonacci numbers is $\Delta = \frac{5^{1/4} \exp\left(-\ln^2(5)/(8\ln\phi)\right) c}{\phi^{1/12}}$, where $c$ is the Fibonacci factorial constant.
• The numerical value of $\Delta$ is approximately 0.8992126807.
• The divergent behavior of $\zeta_F'(s)$ at $s=0$ is resolved by subtracting the principal part $-1/(\ln\phi \cdot s^2)$.
• The constant term in the Laurent expansion of $\zeta_F'(s)$ at $s=0$ is $\frac{1}{24}\left(2\ln\phi + \frac{3\ln^2 5}{\ln\phi} - 6\ln 5\right) - \ln c$.
• The Fibonacci factorial constant $c$ is expressible as $\left(\frac{1}{2}\vartheta_1'\left(0, -i/\phi\right)\right)^{1/3}$, linking the result to modular forms.
• The final expression for $\Delta$ is consistent with analytic continuation and regularization techniques in zeta function theory.