Bernoulli number identities from quantum field theory and topological string theory

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.

Fourier series of sums of products of Genocchi functions and their applications

“…which is the Faber-Pandharipande-Zagier identity (see [4]). Some related works can be found in [3,[12][13][14][15][16].…”

Fourier series of sums of products of Bernoulli functions and their applications

“…Some of the different proofs of Miki's identity can be found in [3,5,14,16]. Dunne-Schubert in [3] uses the asymptotic expansion of some special polynomials coming from the quantum field theory computations, Gessel in [5] is based on two different expressions for Stirling numbers of the second kind S 2 (n, k), Miki in [14] exploits a formula for the Fermat quotient a p −a p modulo p 2 , and Shiratani-Yokoyama in [16] employs p-adic analysis. As we can see, all of these proofs are quite involved.…”

Fourier series of functions involving higher-order ordered Bell polynomials

“…we can derive the Faber-Pandharipande-Zagier identity (see [6]) and the Miki's identity (see [5,17,19]). Some related works on Fourier series expansions for analogous functions can be found in the recent papers [9,10,13,14].…”