5,181 research outputs found
Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers
The Riemann zeta identity at even integers of Lettington, along with his
other Bernoulli and zeta relations, are generalized. Other corresponding
recurrences and determinant relations are illustrated. Another consequence is
the application to sums of double zeta values. A set of identities for the
Ramanujan and generalized Ramanujan polynomials is presented. An alternative
proof of Lettington's identity is provided, together with its generalizations
to the Hurwitz and Lerch zeta functions, hence to Dirichlet series, to
Eisenstein series, and to general Mellin transforms.
The Hurwitz numbers occur in the Laurent expansion about the
origin of a certain Weierstrass function for a square lattice, and are
highly analogous to the Bernoulli numbers. An integral representation of the
Laurent coefficients about the origin for general functions, and for
these numbers in particular, is presented. As a Corollary, the asymptotic form
of the Hurwitz numbers is determined. In addition, a series representation of
the Hurwitz numbers is given, as well as a new recurrence.Comment: 40 pages, no figure
Parameterized summation relations for the Stieltjes constants
The Stieltjes constants appear in the regular part of the
Laurent expansion of the Hurwitz zeta function about its only polar singularity
at . We present multi-parameter summation relations for these constants
that result from identities for the Hurwitz zeta function. We also present
multi-parameter summation relations for functions that may be
expressed as sums over the Stieltjes constants. Integral representations,
especially including Mellin transforms, play an important role. As a byproduct,
reciprocity and other summatory relations for polygamma functions and Bernoulli
polynomials may be obtained.Comment: 15 pages, no figures, 2 new Propositions, Corollaries, and references
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Special functions and the Mellin transforms of Laguerre and Hermite functions
We present explicit expressions for the Mellin transforms of Laguerre and
Hermite functions in terms of a variety of special functions. We show that many
of the properties of the resulting functions, including functional equations
and reciprocity laws, are direct consequences of transformation formulae of
hypergeometric functions. Interest in these results is reinforced by the fact
that polynomial or other factors of the Mellin transforms have zeros only on
the critical line Re s = 1/2. We additionally present a simple-zero Proposition
for the Mellin transform of the wavefunction of the D-dimensional hydrogenic
atom. These results are of interest to several areas including quantum
mechanics and analytic number theory.Comment: 23 pages, no figures, to appear in Analysi
Integral representations of functions and Addison-type series for mathematical constants
We generalize techniques of Addison to a vastly larger context. We obtain
integral representations in terms of the first periodic Bernoulli polynomial
for a number of important special functions including the Lerch zeta,
polylogarithm, Dirichlet - and Clausen functions. These results then enable
a variety of Addison-type series representations of functions. Moreover, we
obtain integral and Addison-type series for a variety of mathematical
constants.Comment: 36 pages, no figure
The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function
The Stieltjes constants gamma_k(a) are the expansion coefficients in the
Laurent series for the Hurwitz zeta function about its only pole at s=1. We
present the relation of gamma_k(1) to the eta_j coefficients that appear in the
Laurent expansion of the logarithmic derivative of the Riemann zeta function
about its pole at s=1. We obtain novel integral representations of the
Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) +
S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the
Riemann hypothesis. The sum S_\gamma(n) is O(n) and we present various integral
representations for it. We present novel series representations of S_2(n). We
additionally present a rapidly convergent expression for \gamma_k= \gamma_k(1)
and a variety of results pertinent to a parameterized representation of the
Riemann and Hurwitz zeta functions.Comment: 37 pages, no figures Prop. 3(b) added and minor update
Generalizations of Russell-style integrals
First some definite integrals of W. H. L. Russell, almost all with
trigonometric function integrands, are derived, and many generalized. Then a
list is given in Russell-style of generalizations of integral identities of
Amdeberhan and Moll. We conclude with a brief and noncomprehensive description
of directions for further investigation, including the significant
generalization to elliptic functions.Comment: 13 pages, no figure
Some definite logarithmic integrals from Euler sums, and other integration results
We present explicit expressions for multi-fold logarithmic integrals that are
equivalent to sums over polygamma functions at integer argument. Such relations
find application in perturbative quantum field theory, quantum chemistry,
analytic number theory, and elsewhere. The analysis includes the use of
properties of a variety of special functions.Comment: 24 pages, no figure
Expected exit times of Brownian motion from planar domains: Complements to a paper of Markowsky
We supplement a very recent paper of G. Markowsky concerned with the expected
exit times of Brownian motion from planar domains. Besides the use of conformal
mapping, we apply results from potential theory. We treat the case of a
wedge-shaped region exactly, subsuming an example of Markowsky. A number of
other results are presented, including for a half disc, -grams, and a
variety of other regions.Comment: 30 pages, 7 figure
Integral and series representations of the digamma and polygamma functions
We obtain a variety of series and integral representations of the digamma
function . These in turn provide representations of the evaluations
at rational argument and for the polygamma function .
The approach is through a limit definition of the zeroth Stieltjes constant
. Several other results are obtained, including product
representations for and for the Gamma function .
In addition, we present series representations in terms of trigonometric
integrals Ci and Si for and the Euler constant .Comment: 27 pages, no figure
Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant
We develop series representations for the Hurwitz and Riemann zeta functions
in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give
the analytic continuation of these functions to the entire complex plane.
Special cases yield series representations of a wide variety of special
functions and numbers, including log Gamma, the digamma, and polygamma
functions. A further byproduct is that values emerge as nonlinear
Euler sums in terms of generalized harmonic numbers. We additionally obtain
series and integral representations of the first Stieltjes constant
. The presentation unifies some earlier results.Comment: 28 pages, no figure
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