9 research outputs found
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 properties of the Lerch's transcendent
A new representation of the Lerch''s transcendent F(z, s, a), valid for positive integer s=n=1, 2, … and for z and a belonging to certain regions of the complex plane, is presented. It allows to write an equation relating F(z, n, a) and F(1/z, n, 1-a), which in turn provides an expansion of F(z, n, a) as a power series of 1/z, convergent for |z|>1
Some discrete Fourier transform pairs associated with the Lipschitz-Lerch Zeta function
It is shown that there exists a companion formula to Srivastavas formula for the Lipschitz-Lerch Zeta function [see H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77-84] and that together these two results form a discrete Fourier transform pair. This Fourier transform pair makes it possible for other (known or new) results involving the values of various Zeta functions at rational arguments to be easily recovered or deduced in a more general context and in a remarkably unified manner. (C) 2009 Elsevier Ltd. All rights reserved
Integral representations of the Legendre chi function
We, by making use of elementary arguments, deduce integral representations of the Legendre chi function chi(S)(z) valid for vertical bar z vertical bar LT 1 and Res GT 1. Our earlier established results on the integral representations for the Riemann zeta function zeta(2n + 1) and the Dirichlet beta function beta(2n), n epsilon N, are a direct consequence of these representations. (c) 2006 Elsevier Inc. All rights reserved
Reflection properties of zeta related functions in terms of fractional derivatives
We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ¿(n, a) of integer first argument can be extended to the more general case of ¿(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ¿(s, a) with (s) > 2 is essentially the Weyl (s-2)-derivative of ¿(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch''s transcendent
Numerical Approximation of Some Infinite Gaussian Series and Integrals
The paper deals with numerical computation of the asymptotic variance of the so-called increment ratio (IR) statistic and its modifications. The IR statistic is useful for estimation and hypothesis testing on fractional parameter H ∈ (0, 1) of random process (time series), see Surgailis et al. [1], Bardet and Surgailis [2]. The asymptotic variance of the IR statistic is given by an infinite integral (or infinite series) of 4-dimensional Gaussian integrals which depend on parameter H. Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series. Graphs and tables of approximate values of the variances σp2(H) and σˆp2(H), p = 1, 2 are included
Special Functions Related to Dedekind Type DC-Sums and their Applications
In this paper we construct trigonometric functions of the sum T_{p}(h,k),
which is called Dedekind type DC-(Dahee and Changhee) sums. We establish
analytic properties of this sum. We find trigonometric representations of this
sum. We prove reciprocity theorem of this sums. Furthermore, we obtain
relations between the Clausen functions, Polylogarithm function, Hurwitz zeta
function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum
T_{p}(h,k). We also give some applications related to these sums and functions