Closed-form evaluations of certain definite integrals by employing the Cauchy integral theorem

It is shown in this paper, by making use of contour integration and the Cauchy integral theorem, that two general families of definite integrals can be evaluated in closed form and are expressible only in terms of the Hurwitz zeta function and elementary functions. In addition, a number of interesting (known or new) special cases and consequences of the main results are considered and some comparison with results of symbolic computation is made.     

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 s=0,−1,−2,…, which are, following Lee, referred to as the polypseudologarithms (or polypseudologs) of order n≡−s. 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.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

Integral representations of the Legendre chi function

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function χs(z) valid for |z|<1 and Res>1. Our earlier established results on the integral representations for the Riemann zeta function ζ(2n+1) and the Dirichlet beta function β(2n), n∈N, are a direct consequence of these representations.     

Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function.     

Summation formulae for finite tangent and secant sums

In a series of papers [6], [7], [8], [9] and [10] it has been shown that nine remarkably general families of the finite trigonometric sums could be summed in closed form by making use of the calculus of residues and choosing a particularly convenient integration contour. In this sequel, new summation formulae for three general families of finite tangent and secant sums have been deduced by the same approach.     

Evaluations of some classes of the trigonometric moment integrals

Four classes of the trigonometric moment integrals are evaluated in closed form in a simple and unified manner by making use of the contour integration in conjunction with the Cauchy integral theorem. In all cases, the closed contour of the same shape is used and it is shown that the integrals are expressible only in terms of the Hurwitz zeta function and elementary functions. A number of interesting (known or new) special cases and consequences of the main results are also considered.     

Values of the Legendre chi and Hurwitz zeta functions at rational arguments

We show that the Hurwitz zeta function, , and the Legendre chi function, , defined by

and

respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.