An elementary approach to the meromorphic continuation of some classical Dirichlet series

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.     

Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

On the spectral expansion of hyperbolic Eisenstein series

In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004), we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L ². Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion.     

Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.     

Poincaré series forSO(n, 1)

A theory of Poincaré series is developed for Lobachevsky space of arbitrary dimension. For a general non-uniform lattice a Selberg-Kloosterman zeta function is introduced. It has meromorphic continuation to the plane with poles at the corresponding automorphic spectrum. When the lattice is a unit group of a rational quadratic form, the Selberg-Kloosterman zeta function is computed explicitly in terms of exponential sums. In this way a non-trivial Ramanujan-like bound analogous to “Selberg’s 3/16 bound” is proved in general.     

Dirichlet series obtained from the error term in the Dirichlet divisor problem

In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term.     

The scattering of a plane wave by a trap in the critical case

The scattering of a plane wave by a resonator with a narrow coupling channel is considered. The velocity potential of the scattered wave in this resonator has two series of poles with small imaginary parts, corresponding to the main trap and the coupling channel, the effect of which inside the trap differs by an order of magnitude. The critical case, when the limiting value for the poles from both series is the same, is investigated. It is shown that in this case two poles exist, which converge to this limiting value, and they both inherit resonance properties, characteristic for poles generated by the main trap. The principal terms of the asymptotic forms of the poles and the scattered wave are constructed.     

An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations

In this paper, a novel single-term Haar wavelet series (STHWS) method is implemented for the solution of the Duffing equation and Painleve’s transcendents (PI and PII). The results, in the form of a block pulse and a discrete solution, are presented. Unlike classical numerical schemes, the STHWS method has no restrictions on the coefficients of the Duffing equation as regards its solution. PI and PII are analysed as regards their solutions, up to nearest singularities (poles), using the STHWS. Also, an efficient computational implementation shows the remarkable features of wavelet based techniques.     

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C¹ smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.     

On Scattering of Poles for Schr?dinger Operator from the Point of View of Dirichlet Series

In this article, we are concerned with the scattering problem of Schr?dinger operators with compactly supported potentials on the real line. We aim at combining the theory of Dirichlet series with scattering theory. New estimate on the number of poles is obtained under the situation that the growth of power series which is related to the potential is not too fast by using a classical result of Littlewood. We propose a new approach of Dirichlet series such that significant upper bounds and lower bounds on the number of poles are obtained. The results obtained in this paper improve and extend some related conclusions on this topic.     

关于薛定谔算子的极点散射: 基于狄利克雷级数的视角

本文讨论了在实轴上具有紧支集的势的薛定谔算子的极点散射问题. 本文旨在将狄利克雷级数理论与散射理论相结合, 文中运用了Littlewood的经典方法得到关于极点个数的新的估计. 本文首次将狄利克雷级数方法用于极点估计, 由此得到了极点个数的上界与下界, 这些结果改进和推广了该论题的一些相关结论.     

Ramanujan and coefficients of meromorphic modular forms

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.     

Splitting of the series of resonances on the nonphysical sheet

The paper is concerned with the behavior of the poles (resonances) of the analytic continuation to the nonphysical sheet of the resolvent of the Dirichlet problem for the operator in the exterior of a trapping domain as the channel connecting the cavity with the exterior of the trap closes up. It is proved that the set of poles decomposes into two series one of which approaches the eigenvalues of the Dirichlet problem in the cavity and the other the resonances of the exterior of the closed trap.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 155–169, 1975.In conclusion, the author wishes to thank B. S. Pavlov for supervision of the work and V. G. Maz'ya and V. A. Solonnikov for useful discussions.     

Note on the differentiability of arithmetic Fourier series arising from Eisenstein series

The aim of this note is to present results concerning the differentiability of some Fourier series arising from Eisenstein series. Sine series exhibit different behaviours with respect to differentiability than the series with cosine function. The precise results are given for the series related to Eisenstein series of weight 2, whereas for the series arising from Eisenstein series of higher weight we conjecture the results.     

Rational approximants to symmetric formal Laurent series

Rational approximants, in the Padé sense, to a given formal Laurent series,F(z)= _– c _k z ^k , have been considered by several authors (see [3] for a survey about the different kinds of approximants which can be defined). In this paper, we shall be concerned with symmetric series, that is, when the complex coefficients {c _k } _– ⁺ satisfyc _–k=c _k,k=0, 1,....Making use of Brezinski's approach [1], for Padé-type approximation to a formal power series, rational approximants toF(z) with prescribed poles are obtained, and their algebraic properties considered. These results will allow us to give an alternative approach for the Padé-Chebyshev approximants.     

A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates

In this paper, we extend our previous work (M.-C. Lai, A simple compact fourth-order Poisson solver on polar geometry, J. Comput. Phys. 182 (2002) 337–345) to 3D cases. More precisely, we present a spectral/finite difference scheme for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. Here the formal fourth-order accuracy means that the scheme is exactly fourth-order accurate while the poles are excluded and is third-order accurate otherwise. Despite the degradation of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condition is needed. The resulting linear system is then solved by the Bi-CGSTAB method with the preconditioner arising from the second-order discretization which shows the scalability with the problem size.     

MODWT-ARMA model for time series prediction

Many time series in the applied sciences display a time-varying second order structure and long-range dependence (LRD). In this paper, we present a hybrid MODWT-ARMA model by combining the maximal overlap discrete wavelet transform (MODWT) and the ARMA model to deal with the non-stationary and LRD time series. We prove theoretically that the details series obtained by MODWT are stationary and short-range dependent (SRD). Then we derive the general form of MODWT-ARMA model. In the experimental study, the daily rainfall and Mackey–Glass time series are used to assess the performance of the hybrid model. Finally, the normalized error comparison with DWT-ARMA, EMD-ARMA and ARIMA model indicates that this combined model is an effective way to improve forecasting accuracy.     

Convergence acceleration of orthogonal series

The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin’s and Weniger’s methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler–Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.      

Time series analysis

Current univariate and multivariate time series modelling procedures are reviewed. Areas of disagreement are discussed, such as the choice of a forecasting method, whether to use time series analysis or econometrics for economic data, the merits of prior seasonal adjustment, and the advisability of prewhitening in multivariate modelling. Some recent developments in model identification, estimation and model criticism are described. It is argued that too little attention has been paid to allowing for departures from the model; this leads to a discussion of exploratory and robust techniques. Finally, it is pointed out that little has been done on modelling positive and discrete-valued time series of importance in OR such as lifetimes and counts; some recent proposals are described.