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Bernoulli Number Identities from Quantum Field Theory and Topological String Theory
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.Comment: 18 pages, final version published in CNTP (title modified, note added
on relevant work that has appeared since the preprint version; otherwise
unchanged
On hypergeometric Bernoulli numbers and polynomials
In this note, we shall provide several properties of hypergeometric Bernoulli
numbers and polynomials, including sums of products identity, differential
equations and recurrence formulas.Comment: 12 page
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