Communications in Number Theory and Physics

Volume 7 (2013)

Number 2

Bernoulli number identities from quantum field theory and topological string theory

Pages: 225 – 249



Gerald V. Dunne (Depts. of Physics and of Mathematics, University of Connecticut, Storrs, Conn., U.S.A.; Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France)

Christian Schubert (Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México)


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.

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