27 research outputs found
Derivative Polynomials and Closed-Form Higher Derivative Formulae
In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for
rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed
form symbolic derivatives of four functions belonging to the class of functions
whose derivatives are polynomials in terms of the same functions. In this
sequel, simple closed-form higher derivative formulae which involve the
Carlitz-Scoville higher order tangent and secant numbers are derived for eight
trigonometric and hyperbolic functions.Comment: 7 page
Closed-form formulae for the derivatives of trigonometric functions at rational multiples of
In this sequel to our recent note it is shown, in a unified manner, by making
use of some basic properties of certain special functions, such as the Hurwitz
zeta function, Lerch zeta function and Legendre chi function, that the values
of all derivatives of four trigonometric functions at rational multiples of
can be expressed in closed form as simple finite sums involving the
Bernoulli and Euler polynomials. In addition, some particular cases are
considered.Comment: 5 page
New integral representations of the polylogarithm function
Maximon has recently given an excellent summary of the properties of the
Euler dilogarithm function and the frequently used generalizations of the
dilogarithm, the most important among them being the polylogarithm function
. The polylogarithm function appears in several fields of mathematics
and in many physical problems. We, by making use of elementary arguments,
deduce several new integral representations of the polylogarithm for any
complex z for which < 1. Two are valid for all complex s, whenever
. The other two involve the Bernoulli polynomials and are valid in
the important special case where the parameter s is an positive integer. Our
earlier established results on the integral representations for the Riemann
zeta function ,, follow directly as corollaries
of these representations.Comment: 15 page
A dilogarithmic integral arising in quantum field theory
Recently, an interesting dilogarithmic integral arising in quantum field
theory has been closed-form evaluated in terms of the Clausen function
by Coffey [J. Math. Phys.} 49 (2008), 093508]. It
represents the volume of an ideal tetrahedron in hyperbolic space and is
involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It
is shown here, by simple and direct arguments, that this integral can be
expressed by the triplet of the Clausen function values which are involved in
one of the two above-mentioned conjectures.Comment: 6 page
Integral representations of the Riemann zeta function for odd-integer arguments
AbstractWe deduce four new integral representations for ζ(2n+1),n∈N, where ζ(s) is the Riemann zeta function
Polypseudologarithms revisited
Lee, in a series of papers, described a unified formulation of the
statistical thermodynamics of ideal quantum gases in terms of the polylogarithm
functions, . It is aimed here to investigate the functions
for which are, following Lee,
referred to as the polypseudologarithms (or polypseudologs) of order .
Various known results regarding polypseudologs, mainly obtained in widely
differing contexts and currently scattered throughout the literature, have been
brought together along with many new results and insights and they all have
been proved in a simple and unified manner. In addition, a new general explicit
closed-form formula for these functions involving the Carlitz--Scoville higher
tangent numbers has been established.Comment: 10 page