789 research outputs found
Commutation relations of operator monomials
In this short paper, the commutator of monomials of operators obeying
constant commutation relations is expressed in terms of anticommutators. The
formula involves Bernoulli numbers or Euler polynomials evaluated in zero. The
role of Bernoulli numbers in quantum-mechanical identities such as the
Baker-Campbell-Hausdorff formula is emphasized and applications connected to
ordering problems as well as to Ehrenfest theorem are proposed.Comment: submitted to J. Phys. A: Math. Theo
A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers
We connect and generalize Matiyasevich's identity #0102 with Bernoulli
numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan
summation of the divergent series of the infinite sum of the harmonic numbers.
The formulae are analytic continuation of Euler sums and lead to new recursion
relations for derivatives of Bernoulli numbers. The techniques used are contour
integration, generating functions and divergent series.Comment: 10 page
On Plouffe's Ramanujan Identities
Recently, Simon Plouffe has discovered a number of identities for the Riemann
zeta function at odd integer values. These identities are obtained numerically
and are inspired by a prototypical series for Apery's constant given by
Ramanujan: Such sums follow from a general relation given by Ramanujan, which is
rediscovered and proved here using complex analytic techniques. The general
relation is used to derive many of Plouffe's identities as corollaries. The
resemblance of the general relation to the structure of theta functions and
modular forms is briefly sketched.Comment: 19 pages, 3 figures; v4: minor corrections; modified intro; revised
concluding statement
A Quantum Field Theoretical Representation of Euler-Zagier Sums
We establish a novel representation of arbitrary Euler-Zagier sums in terms
of weighted vacuum graphs. This representation uses a toy quantum field theory
with infinitely many propagators and interaction vertices. The propagators
involve Bernoulli polynomials and Clausen functions to arbitrary orders. The
Feynman integrals of this model can be decomposed in terms of an algebra of
elementary vertex integrals whose structure we investigate. We derive a large
class of relations between multiple zeta values, of arbitrary lengths and
weights, using only a certain set of graphical manipulations on Feynman
diagrams. Further uses and possible generalizations of the model are pointed
out.Comment: Standard latex, 31 pages, 13 figures, final published versio
Expansions of generalized Euler's constants into the series of polynomials in and into the formal enveloping series with rational coefficients only
In this work, two new series expansions for generalized Euler's constants
(Stieltjes constants) are obtained. The first expansion involves
Stirling numbers of the first kind, contains polynomials in with
rational coefficients and converges slightly better than Euler's series . The second expansion is a semi-convergent series with rational
coefficients only. This expansion is particularly simple and involves Bernoulli
numbers with a non-linear combination of generalized harmonic numbers. It also
permits to derive an interesting estimation for generalized Euler's constants,
which is more accurate than several well-known estimations. Finally, in
Appendix A, the reader will also find two simple integral definitions for the
Stirling numbers of the first kind, as well an upper bound for them.Comment: Copy of the final journal version of the pape
On asymptotics, Stirling numbers, Gamma function and polylogs
We apply the Euler--Maclaurin formula to find the asymptotic expansion of the
sums , ~, ~, ~ in closed form to arbitrary order (). The expressions often simplify considerably and the coefficients are
recognizable constants. The constant terms of the asymptotics are either
(first two sums), 0 (third sum) or yield novel
mathematical constants (fourth sum). This allows numerical computation of
faster than any current software. One of the constants
also appears in the expansion of the function
around the singularity at ; this requires the asymptotics of the
incomplete gamma function. The manipulations involve polylogs for which we find
a representation in terms of Nielsen integrals, as well as mysterious
conjectures for Bernoulli numbers. Applications include the determination of
the asymptotic growth of the Taylor coefficients of . We also
give the asymptotics of Stirling numbers of first kind and their formula in
terms of harmonic numbers.Comment: 24 pages, to appear in Results for Mathematic
Asymptotic expansions of exponentials of digamma function and identity for Bernoulli polynomials
The asymptotic expansion of digamma function is a starting point for the
derivation of approximants for harmonic sums or Euler-Mascheroni constant. It
is usual to derive such approximations as values of logarithmic function, which
leads to the expansion of the exponentials of digamma function. In this paper
the asymptotic expansion of the function is derived and
analyzed in details, especially for integer values of parameter . The
behavior for integer values of is proved and as a consequence a new
identity for Bernoulli polynomials. The obtained formulas are used to improve
know inequalities for Euler's constant and harmonic numbers
On the behavior of -adic Euler -functions
In this paper we propose a construction of -adic Euler -function
using Kubota-Leopoldt's approach and Washington's one. We also compute the
derivative of -adic Euler -function at and the values of
-adic Euler -function at positives integers.Comment: 16 page
The Euler Legacy to Modern Physics
Particular families of special functions, conceived as purely mathematical
devices between the end of XVIII and the beginning of XIX centuries, have
played a crucial role in the development of many aspects of modern Physics.
This is indeed the case of the Euler gamma function, which has been one of the
key elements paving the way to string theories, furthermore the Euler-Riemann
Zeta function has played a decisive role in the development of renormalization
theories. The ideas of Euler and later those of Riemann, Ramanujan and of
other, less popular, mathematicians have therefore provided the mathematical
apparatus ideally suited to explore, and eventually solve, problems of
fundamental importance in modern Physics. The mathematical foundations of the
theory of renormalization trace back to the work on divergent series by Euler
and by mathematicians of two centuries ago. Feynman, Dyson, Schwinger...
rediscovered most of these mathematical "curiosities" and were able to develop
a new and powerful way of looking at physical phenomena
On Jacobi and continuous Hahn polynomials
Jacobi polynomials are mapped onto the continuous Hahn polynomials by the
Fourier transform and the orthogonality relations for the continuous Hahn
polynomials then follow from the orthogonality relations for the Jacobi
polynomials and the Parseval formula. In a special case this relation dates
back to work by Bateman in 1933 and we follow a part of the historical
development for these polynomials. Some applications of this relation are
given
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