1,070 research outputs found
On a class of -Bernoulli, -Euler and -Genocchi polynomials
The main purpose of this paper is to introduce and investigate a class of
-Bernoulli, -Euler and -Genocchi polynomials. The -analogues of
well-known formulas are derived. The -analogue of the Srivastava--Pint\'er
addition theorem is obtained. Some new identities involving -polynomials are
proved
A Few Finite Trigonometric Sums
Finite trigonometric sums occur in various branches of physics, mathematics,
and their applications. These sums may contain various powers of one or more
trigonometric functions. Sums with one trigonometric function are known,
however sums with products of trigonometric functions can get complicated and
may not have a simple expressions in a number of cases. Some of these sums have
interesting properties and can have amazingly simple value. However, only some
of them are available in literature. We obtain a number of such sums using
method of residues.Comment: 11 pages, added references, corrected typo
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
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