[Paper Review] Bernoulli Number Identities from Quantum Field Theory

Gerald V. Dunne, Christian Schubert|arXiv (Cornell University)|Jun 29, 2004

Advanced Mathematical Identities19 references15 citations

TL;DR

This paper introduces a novel quantum field theory-inspired method to derive convolution identities for Bernoulli numbers, unifying and generalizing known identities like Miki's and the Faber-Pandharipande-Zagier identity into continuous-parameter families. It yields a new Bernoulli number convolution identity and extends the framework to Euler-Bernoulli convolutions, offering a systematic approach to higher-order identities.

ABSTRACT

We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families of convolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki's identity due to Gessel and obtain a new similar identity generalizing the FPZ identity. The generalization of the method to the derivation of convolution identities of arbitrary order is outlined. We also describe an extension to identities which relate convolutions of Euler and Bernoulli numbers.

Motivation & Objective

• To develop a unified, physics-motivated approach for deriving convolution identities of Bernoulli numbers.
• To reveal the structural relationship between the Miki and Faber-Pandharipande-Zagier identities through a common framework.
• To generalize known identities into continuous-parameter families, enabling broader applicability.
• To extend the method to higher-order convolution identities and to identities involving Euler numbers.
• To provide a systematic derivation method rooted in quantum field theory and string theory principles.

Proposed method

• Leverage quantum field theory and string theory insights to motivate the structure of Bernoulli number convolution identities.
• Use a unified algebraic framework to derive the Miki and FPZ identities simultaneously, revealing their underlying connection.
• Introduce a continuous parameter to generalize each of the three identities—Miki’s, FPZ’s, and the newly derived identity—into infinite families.
• Apply the method to re-derive Gessel’s cubic generalization of Miki’s identity, confirming consistency and robustness.
• Outline a generalization to arbitrary-order convolution identities using the same underlying principle.
• Extend the formalism to identities involving both Euler and Bernoulli numbers, broadening its mathematical scope.

Experimental results

Research questions

• RQ1 How are the Miki and Faber-Pandharipande-Zagier identities structurally related, and can they be derived from a common principle?
• RQ2 Can the Miki and FPZ identities be generalized into continuous-parameter families of convolution identities?
• RQ3 What new convolution identities emerge from the proposed method, and how do they relate to known results?
• RQ4 Can the method be extended to higher-order convolution identities beyond quadratic forms?
• RQ5 Can the framework be adapted to derive identities involving both Euler and Bernoulli numbers?

Key findings

• A new Bernoulli number convolution identity is derived through a unified approach that naturally unifies the Miki and FPZ identities.
• The method successfully generalizes both the Miki and FPZ identities into continuous-parameter families, revealing deeper structural patterns.
• The framework rederives Gessel’s cubic generalization of Miki’s identity, validating its consistency and power.
• A new cubic identity generalizing the FPZ identity is discovered, extending its known form.
• The approach is systematically extendable to arbitrary-order convolution identities, suggesting a broad mathematical framework.
• The method is generalized to include identities relating convolutions of Euler and Bernoulli numbers, broadening its applicability.