Plana Summation Formula Research Articles

Orthogonal polynomials with respect to the Abel weight

While a lot is known about the classical orthogonal polynomials, their counterparts with respect to non-classical weights are not as well explored. Nevertheless, sometimes such weights come handy as well. For example, the famous Abel–Plana summation formula offers a convenient method of summing an infinite series, reducing the sum to an integral with the Abel weight function on the real line, w(x)=x2sinhπx. Orthogonal polynomials with respect to this weight naturally arise when we have to numerically evaluate this integral using the Gauss quadrature rule. These orthogonal polynomials are the object of study in this paper. We obtain a number of explicit formulas and algebraic relations between these and related polynomials, including the associated polynomials. In particular, for many of these polynomials we obtain Fourier expansions with the orthogonal polynomials as the basis. We also determine the weight functions whose orthogonal polynomials are the polynomials we discussed. At the end, we briefly discuss the asymptotic and perform numerical experiments.

A Unifying Principle in the Theory of Modular Relations

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function Z(s) satisfying a suitable functional equation, one can generalize it to Zf(s) in the form of an integral involving the Mellin transform F(s) of a certain suitable function f(x) and process it further as Z˜f(s). Under the condition that F(s) is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for Z˜f(s). Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion).

Integral representations for certain series

In this work we apply the classical Plana summation formula to derive integral representations for certain infinite series that their general terms containing Gamma, q q -Gamma, theta, Bessel, and Hurwitz zeta functions.

The Euler–Maclaurin–Siegel and Abel–Plana summation formulae for the entire Riemann functional equation to handle the Riemann hypothesis

In this article, with the aid of the entire Riemann functional equation (ERFE), defined by ξs=12ss−1π−s2Γs2ςs, where s is a complex variable, Γs is the Euler’s gamma function, and ςs is the Riemann Zeta function (RZF), the Euler–Maclaurin–Siegel summation formula (EMSSF) and the Abel–Plana summation formula (APSF) are addressed to prove the Riemann hypothesis (RH) for the first time. The theorems for the ERFE are presented, and the complex zeros for the ERFE and RZF are also discussed in detail. The presented results are accurately and efficiently proposed to find the critical line of the ERFE.

Thermal corrections to the Casimir energy in a Lorentz-breaking scalar field theory

In this paper, we investigate the thermal effect on the Casimir energy associated with a massive scalar quantum field confined between two large parallel plates in a CPT-even, aether-like Lorentz-breaking scalar field theory. In order to do that, we consider a nonzero chemical potential for the scalar field assumed to be in thermal equilibrium at some finite temperature. The calculations of the energies are developed by using the Abel–Plana summation formula, and the corresponding results are analyzed in several asymptotic regimes of the parameters of the system, like mass, separations between the plates and temperature.

Induced fermionic currents in de Sitter spacetime in the presence of a compactified cosmic string

We investigate the vacuum fermionic currents in the geometry of a compactified cosmic string on the background of de Sitter (dS) spacetime. The currents are induced by magnetic fluxes running along the cosmic string and enclosed by the compact dimension. We show that the vacuum charge and the radial component of the current density vanish. By using the Abel–Plana summation formula, the azimuthal and axial currents are explicitly decomposed into two parts: the first one corresponds to the geometry of a straight cosmic string, and the second one is induced by the compactification of the string along its axis. For the axial current the first part vanishes and the corresponding topological part is an even periodic function of the magnetic flux along the string axis and an odd periodic function of the flux enclosed by the compact dimension with the periods equal to the flux quantum. The azimuthal current density is an odd periodic function of the flux along the string axis and an even periodic function of the flux enclosed by the compact dimension with the same period. Depending on the magnetic fluxes, the planar angle deficit can either enhance or reduce the azimuthal and axial currents. The influence of the background gravitational field on the vacuum currents is crucial at distances from the string larger than the dS curvature radius. In particular, for the geometry of a straight cosmic string and for a massive fermionic field, we show that the decay of the azimuthal current density is damping oscillatory with the amplitude inversely proportional to the fourth power of the distance from the string. This behavior is in clear contrast with the case of the string in Minkowski bulk, where the current density is exponentially suppressed at large distances.

Plana's summation formula as a modular relation and applications

In this paper, we shall locate Plana's summation formula à la Koshlyakov as a form of the modular relation for the Riemann zeta-function and an analytic continuation technique furnished by Lemma 1. Then by Plana's summation formula, we shall prove an important integral representation for the Hurwitz–Lerch zeta-function through confluent hypergeometric functions, which cover a wide range of integral representations attributed to famous mathematicians. We shall also locate the genesis of Mikolás’ formula and prove that the functional equations for the Hurwitz zeta-function and the Riemann zeta-function are equivalent.

Finite-temperature Casimir effect in Randall–Sundrum models

The finite-temperature Casimir effect for a scalar field in the bulk region of two Randall–Sundrum models, RSI and RSII, is studied. We calculate the Casimir energy and the Casimir force for two parallel plates with separation a on the visible brane in the RSI model. High-temperature and low-temperature cases are covered. Attraction versus repulsion of the temperature correction to the force is discussed in the special cases of Dirichlet–Dirichlet, Neumann–Neumann and Dirichlet–Neumann boundary conditions at low temperature. The Abel–Plana summation formula is used, as this is found to be the most convenient. Some comments are made on the related contemporary literature.

Casimir effect in de Sitter space–time with compactified dimension

We investigate the Hadamard function, the vacuum expectation values of the field square and the energy–momentum tensor of a scalar field with general curvature coupling parameter in de Sitter space–time compactified along one of spatial dimensions. By using the Abel–Plana summation formula, we have explicitly extracted from the vacuum expectation values the part due to the compactness of the spatial dimension. The topological part in the vacuum energy–momentum tensor violates the local de Sitter symmetry and dominates in the early stages of the cosmological evolution. At late times the corresponding vacuum stresses are isotropic and the topological part corresponds to an effective gravitational source with barotropic equation of state.

Spinor Casimir densities for a spherical shell in the global monopole spacetime

We investigate the vacuum expectation values of the energy–momentum tensor and the fermionic condensate associated with a massive spinor field obeying the MIT bag boundary condition on a spherical shell in the global monopole spacetime. In order to do that, we use the generalized Abel–Plana summation formula. As we shall see, this procedure allows us to extract from the vacuum expectation values the contribution coming from the unbounded spacetime and to explicitly present the boundary induced parts. As regards the boundary induced contribution, two distinct situations are examined: the vacuum average effects inside and outside the spherical shell. The asymptotic behaviour of the vacuum densities is investigated near the sphere centre and near the surface, and at large distances from the sphere. In the limit of strong gravitational field corresponding to small values of the parameter describing the solid angle deficit in the global monopole geometry, the sphere induced expectation values are exponentially suppressed. We discuss, as a special case, the fermionic vacuum densities for the spherical shell on the background of the Minkowski spacetime. Previous approaches to this problem within the framework of the QCD bag models have been global and our calculation is a local extension of these contributions.

Casimir effect of the Maxwell-Chern-Simons field for two non-parallel lines boundary

Based on the Faddeev formalism of path-integral quantization for a constrained Hamiltonian system, the Casimir effect between two non-parallel lines in the (2+1)-dimensional space is calculated by using conformal mapping and Plana summation formula in the theory of complex variable function. Without introducing any cutoff of parameter, the finite analytical expression is obtained.

On the Numerical Evaluation of Legendre's Chi-Function

Legendre's chi-function, [formula] , is reexpanded in a power series in powers of [formula] . The expansion obtained is well suited for the computation of [formula] in the two cases of real z close to 1, and [formula] . For n=2 and n=3 , the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses Plana's summation formula.

On Evaluation of Moments of K ν (t)/I ν (t)

This paper presents a method of evaluation of the moments of K,(t)/l,(t).iTwo pairs of expressions, each consisting of two series, are obtained according to the index being an even or an odd integer. The method is an extension of the method used by Watson. Values are tabulated to 12D for p = 0(1)2. In a recent paper [1], Roberts considered the computation of the integral (1 ) f dtk J dt (k _ 2v), where I, and K, are modified Bessel functions. He developed the integral into an 'asymptotic series' suitable for computation when k is a large integer. When k is otherwise a small integer, he integrated numerically the following equivalent integral by using Simpson's rule: (2) = k411i(t)dt (k > 2v). In particular, values are tabulated to 8S for v = 1 and k = 2(1)100. Some time ago, Watson [2] evaluated the integral in (2) when k is an even integer. He developed the integral into two series by employing a method based on a modification of Plana's summation formula. It is found that this method can be extended to the case when k is an odd integer. Furthermore, it is also found that the same integral can be developed into different series by a modification of the method. Altogether, two pairs of expressions are obtained according to k being an even or an odd integer. It is the purpose of this paper to present such results. Values of the integral are thereby evaluated to 12D for v' = 0(1)2. It is mentioned that a method analogous to the present one was used recently by the authors for the evaluation of two Howland integrals [3]. For convenience, the integral is redefined, together with a factor, as follows: k+1 (3) ~~~~~~2 kK,(t)_ 7r(k!) lo I,(t) ( v so that it tends asymptotically to unity as k tends to infinity. The equivalent integral is 2k+l1 C tk (4) L 1)! f kdt (k > 2v). 7r(k + 1)! 1I(V The following results are obtained for k > v: Received September 22, 1971. AMS 1969 subject classifications. Primary 6525.