109 research outputs found
Harmonic motion and Cassini ovals
We consider a two-dimensional free harmonic oscillator where the initial
position is fixed and the initial velocity can change direction. All possible
orbits are ellipses and their enveloping curve is an ellipse too. We show that
the locus of the foci of all elliptical orbits is a Cassini oval. Depending on
the magnitude of the initial velocity we observe all three kinds of Cassini
ovals, one of which is the lemniscate of Bernoulli. These Cassini ovals have
the same foci as the enveloping ellipse.Comment: Eight figures created with GeoGebra. The paper has been presented in
a talk at Ohio Northern University in September 201
Evaluation of one exotic Furdui type series
This is a closed form evaluation of one interesting alternating series whose
value is a combination of three mathematical constants, Pi, log 2, and Zeta
(3).Comment: Updated version of the paper in the Far East Journal of Mathematical
Science
A special constant and series with zeta values and harmonic numbers
In this paper we demonstrate the importance of a mathematical constant which
is the value of several interesting numerical series involving harmonic
numbers, zeta values, and logarithms. We also evaluate in closed form a number
of numerical and power series.Comment: 19 page
Derivative Polynomials for tanh, tan, sech and sec in Explicit Form
The derivative polynomials for the hyperbolic and trigonometric tangent,
cotangent and secant are found in explicit form, where the coefficients are
given in terms of Stirling numbers of the second kind. As application, some
integrals are evaluated and the reflection formula for the polygamma function
is written in explicit form.Comment: A similar version in The Fibonacci Quarterly, 200
Evaluation of some simple Euler-type series
Five series are evaluated in terms of zeta values. Three of the series
involve harmonic numbers and one involves Stirling numbers of the first kind.
The evaluation of these series is reduced to the evaluation of certain
integrals, including the moments of the polylogarithm
Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials
This is a short survey of a class of functions introduces by Tom Apostol. The
survey is focused on their relation to Eulerian polynomials, derivative
polynomials, and also on some integral representations
On a series of Furdui and Qin and some related integrals
In this note we present the solution of Problem H-691 (The Fibonacci
Quarterly, 50 (1) 2012) with some corrections and more details. The solution
involves three nontrivial integrals whose evaluations are given here
Eratosthenes and Pliny, Greek geometry and Roman follies
Supportive attitudes can bring to a blossoming science, while neglect can
quickly make science absent from everyday life and provide a very primitive
view of the world. We compare one important Greek achievement, the computation
of the Earth meridian by Eratosthenes, to its later interpretation by the Roman
historian of science Pliny.Comment: 9 page
Binomial transform and the backward difference
We prove an important property of the binomial transform: it converts
multiplication by the discrete variable into a certain difference operator. We
also consider the case of dividing by the discrete variable. The properties
presented here are used to compute various binomial transform formulas
involving harmonic numbers, skew-harmonic numbers, Fibonacci numbers, and
Stirling numbers of the second kind. Several new identities are proved and some
known results are given new short proofs.Comment: The paper is a slight modification of the journal article in Advances
and Applications in Discrete Mathematics, 13 (1) (2014), 43-6
Euler Sums of Hyperharmonic Numbers
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical
harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers:
{\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in
terms of series of Hurwitz zeta function values. This is a generalization of a
result of Mez\H{o} and Dil. We also provide an explicit evaluation of
{\sigma}(r,m) in a closed form in terms of zeta values and Stirling numbers of
the first kind. Furthermore, we evaluate several other series involving
hyperharmonic numbers.Comment: 9 page
- …