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Bernoulli number identities from quantum field theory and topological string theory. (English) Zbl 1297.11009
Authors’ 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 [H. Miki, J. Number Theory 10, 297–302 (1978; Zbl 0379.10007)] and the Faber-Pandharipande-Zagier (FPZ) identity [C. Faber and R. Pandharipande, Invent. Math. 139, 173–199 (2000; Zbl 0960.14031)] 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 I. M. Gessel [J. Number Theory 110, No. 1, 75–82 (2005; Zbl 1073.11013)] 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.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
05A19 Combinatorial identities, bijective combinatorics
81T10 Model quantum field theories
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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