Coefficients Of Power Series Expansion Research Articles

Calculation of multipole moments of axistationary electrovacuum spacetimes

The multipole moments of stationary axially symmetric vacuum or electrovacuum spacetimes can be expressed in terms of the power series expansion coefficients of the Ernst potential on the axis. In this paper we present a simpler, more efficient calculation of the multipole moments, applying methods introduced by B\"ackdahl and Herberthson. For the nonvacuum electromagnetic case, our results for the octupole and higher moments differ from the results already published in the literature. The reason for this difference is that we correct an earlier unnoticed mistake in the power series solution of the Ernst equations. We also apply the presented method to directly calculate the multipole moments of a five-parameter charged magnetized generalization of the Kerr and Tomimatsu-Sato exact solutions.

A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations

We review the series solutions of the general and single-confluent Heun equations in terms of powers, ordinary-hypergeometric and confluent-hypergeometric functions. The conditions under which the expansions reduce to finite sums as well as the cases when the coefficients of power-series expansions obey two-term recurrence relations are discussed. Infinitely many cases for which a solution of a general or single-confluent Heun equation is written through a single generalized hypergeometric function are indicated. It is shown that the parameters of this generalized hypergeometric function are the roots of a certain polynomial equation, which we explicitly present for any given order.

Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems: Simple eigenvalues

The study of the vibrational modes and stability of a given physical system is strongly tied to the efficient numerical evaluation of its eigenvalues. The operators governing the eigenproblem are, in general, nonlinear in the eigenvalue and non-self-adjoint, which makes the repeated solution of the eigenvalue problem (necessary, for example, when the effect of several parameter values on the system needs to be assessed) expensive. This study reviews the adjoint-based incremental procedure for calculating the coefficients of power series expansions of simple (non-degenerate) eigenvalues and their eigenvectors. These expansions approximate the eigenvalues to any desired order in a finite region. An efficient numerical implementation of the theory is proposed, and it is shown how high-order power series approximations of the eigenvalues give very accurate results within the radius of convergence of the power series, which is finite and generally not small. Furthermore, the domain of convergence of the power series might be extended by considering Padé expansions of the eigenvalues. Examples involving the stability of the Orr–Sommerfeld equation, the biharmonic equation for the vibrational modes of a membrane, and the emission of sound from a Rijke tube, associated with thermoacoustic feedback, are used to assess and validate the theory.

Monotonicity and positivity of coefficients of power series expansions associated with Newton and Halley methods for the matrix pth root

If A is a matrix with all eigenvalues in the disk |z−1|<1, the principal pth root of A can be computed by Newton's method or Halley's method. The study of Newton's method and Halley's method for the matrix pth root can be done through a study of power series expansions of some sequences of scalar functions. In this paper, we prove monotonicity results for the coefficients in the power series expansions for both Newton's method and Halley's method, and for all integer p≥2. The sign patterns of these coefficients can be seen directly from the monotonicity results. We then use these monotonicity results and their consequences to obtain some new error estimates in the matrix case, to obtain a monotonic convergence result when A is a nonsingular M-matrix, and to obtain a structure preserving result when A is a nonsingular M-matrix or a real nonsingular H-matrix with positive diagonal entries.

Arithmetic properties of coefficients of power series expansion of prod _{n=0}^{infty }left( 1-x^{2^{n}}right) ^{t} (with an appendix by Andrzej Schinzel)

Let F(x)=prod _{n=0}^{infty }(1-x^{2^{n}}) be the generating function for the Prouhet–Thue–Morse sequence ((-1)^{s_{2}(n)})_{nin {mathbb {N}}}. In this paper we initiate the study of the arithmetic properties of coefficients of the power series expansions of the function Ft(x)=F(x)t=∑n=0∞fn(t)xn.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} F_{t}(x)=F(x)^{t}=\\sum _{n=0}^{\\infty }f_{n}(t)x^{n}. \\end{aligned}$$\\end{document}For tin {mathbb {N}}_{+} the sequence (f_{n}(t))_{nin {mathbb {N}}} is the Cauchy convolution of t copies of the Prouhet–Thue–Morse sequence. For tin {mathbb {Z}}_{<0} the n-th term of the sequence (f_{n}(t))_{nin {mathbb {N}}} counts the number of representations of the number n as a sum of powers of 2 where each summand can have one among -t colors. Among other things, we present a characterization of the solutions of the equations f_{n}(2^k)=0, where kin {mathbb {N}}, and f_{n}(3)=0. Next, we present the exact value of the 2-adic valuation of the number f_{n}(1-2^{m})—a result which generalizes the well known expression concerning the 2-adic valuation of the values of the binary partition function introduced by Euler and studied by Churchhouse and others.

Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.

One loop integration with hypergeometric series by using recursion relations

General one-loop integrals with arbitrary mass and kinematical parameters in d-dimensional space-time are studied. By using Bernstein theorem, a recursion relation is obtained which connects (n + 1)-point to n-point functions. In solving this recursion relation, we have shown that one-loop integrals are expressed by a newly defined hypergeometric function, which is a special case of Aomoto-Gelfand hypergeometric functions.We have also obtained coefficients of power series expansion around 4-dimensional space-time for two-, three- and four-point functions. The numerical results are compared with "LoopTools" for the case of two- and three-point functions as examples.

CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

Congruences via modular forms

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apery-like differential equations.

Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

Congruence Relations Connecting Tate-Shafarevich Groups with Hurwitz Numbers

Let p > 3 be a rational prime congruent to 3 modulo 4, and h(−p) be the class number of the imaginary quadratic field Q(\\sqrt{-p}). Then h(−p)≡−2B\\frac{p+1}{2}modp, where Bn is the n-th Bernoulli number. This is a quite classical congruence. Under the full BSD conjecture, we provide an easy method to obtain the natural explicit generalization of this, which is a congruence between the conjectural order of the Tate-Shafarevich group for certain elliptic curve with Mordell-Weil rank 0 and a coefficient of power series expansion of an elliptic function associating the elliptic curve.

Exact solutions to sourceless charged massive scalar field equation on Kerr–Newman background

The covariant Klein–Gordon equation in the Kerr–Newman black hole geometry is separated into a radial part and an angular part. It is discovered that in the nonextreme case, these two equations belong to a generalized spin-weighted spheroidal wave equation. Then general exact solutions in integral forms and several special solutions with physical interest are given. While in the extreme case, the radial equation can be transformed into a generalized Whittaker–Hill equation. In both cases, five-term recurrence relations between coefficients in power series expansion of general solutions are presented. Finally, the connection between the radial equations in both cases is discussed.

An FFT-based algorithm for 2D power series expansions

An effective numerical algorithm based on inverting a specialized Laplace transform is derived for computing the two-dimensional power-series expansion coefficients of a two-variable function. Due to the special structure of the constructed 2D Laplace transform, the accuracy of the inverted function values can be assured effectively by the generalized Riemann zeta function evaluation and the multiple sets of 2D FFT computation. Therefore, the algorithm is particularly amenable to modern computers having multiprocessors and/or vector processors.

Routh-Pade approximation for interval systems

Presents a method for the reduction of the order of interval system. The denominator of the reduced model is obtained by a direct truncation of the Routh table of the interval system. The numerator is obtained by matching the coefficients of power series expansions of the interval system and its reduced model. A numerical example illustrates the proposed procedure. >

Representation of the power-series expansion coefficients for the one-point correlation function in the grand canonical ensemble

A representation is found for the coefficients of the expansion of the one-point correlation function (the one-particle distribution density) in a series in powers of the activity that makes it possible to calculate, at least approximately, the first few coefficients of the expansion. The results can also be used to investigate problems of the thermodynamic limit in the grand canonical ensemble.

Power series expansions of Riemann’s $\xi$ function

We show how high-precision values of the coefficients of power series expansions of functions related to Riemann’s $\xi$, function may be calculated. We also show how the Stieltjes constants can be evaluated using this scheme and how the Riemann hypothesis can be expressed in terms of the behavior of two of the sequences of coefficients. High-precision values for the coefficients of these power series are found using Mathematica extsctm.

Calculating the energy parameters for the CV and OPF methods

AbstractA method is proposed for calculating the energies ε (with ε being the energy of the m‐configuration of a type s figure), which are parameters of the cluster variation (CV) method and the order parameter functional (OPF) method. An analytic expression is derived which describes the energy ε in terms of power series expansion coefficients of the formation enthalpy of a disordered crystal. It is shown that a necessary condition for ordering is the presence of a nonlinear composition dependence of the formation enthalpy of a disordered crystal.

Derivation of continued-fraction expansions from the coefficients of power-series expansions of a transfer function

This correspondence presents a rule for deriving various forms of continued-fraction expansions of a transfer function from its power-series expansions around several points. This rule is especially useful for computing multipoint Padé approximants to a given function rational or irrational.

Retrieval of interaction coefficients from total vapor pressure-composition data

A numerical-derivative method, employing a running cubic spline, has been developed to infer coefficients in power series expansions of total vapor pressure vs. composition data. The technique makes it possible to determine values of the coefficients individually, avoiding a generic problem in least squares analysis, viz., that the regression parameters are highly coupled and that their derived values depend strongly on the number of constants used in fitting data. The numerical derivative method also indicates when errors of measurements will preclude the determination of meaningful values of higher-order terms in polynomical representations of data.

Pressure-Induced Tricritical Nature in Triglycine Selenate

Temperature dependences of dielectric constant and spontaneous polarization in triglycine selenate were measured under hydrostatic pressure. Coefficients of power series expansion of the free energy in respect to electric polarization were calculated front the experimental data. It has been found that the coefficient ξ of the fourth power changes its sign from plus to minus near 5 kbar. This indicates that the tricritical point of triglycine selenate exists at ∼40.6°C and ∼5 kbar.