Series Expansion Formulas Research Articles

New formulas for Maclaurin's series expansions of powers of arctangent and other related topics

The work proposes new formulas for Maclaurin's series expansions of natural and real powers of the arctangent and hyperbolic arctangent, a new explicit formula for the Meixner polynomial of the second kind, an explicit expression for the coefficients associated with the moment of neutron density of a point source in an infinitely scattering medium, an explicit formula for the coefficients that are related to the series expansion proposed by Rzadkowskii of the Riemann zeta function, as well as several other identities containing Bell polynomials of the second kind and Stirling numbers of the first kind.

A series expansion formula of the scale matrix with applications in CUSUM analysis

We introduce a new Lévy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.

Probabilistic Network Topology Prediction for Active Planning: An Adaptive Algorithm and Application

This article tackles the problem of active planning to achieve cooperative localization for multirobot systems under measurement uncertainty in GNSS-limited scenarios. Specifically, we address the issue of accurately predicting the probability of a future connection between two robots equipped with range-based measurement devices. Due to the limited range of the equipped sensors, edges in the network connection topology will be created or destroyed as the robots move with respect to one another. Accurately predicting the future existence of an edge, given imperfect state estimation and noisy actuation, is therefore a challenging task. An adaptive power series expansion (or APSE) algorithm is developed based on current estimates and control candidates. Such an algorithm applies the power series expansion formula of the quadratic positive form in a normal distribution. Finite-term approximation is made to realize the computational tractability. Further analyses are presented to show that the truncation error in the finite-term approximation can be theoretically reduced to a desired threshold by adaptively choosing the summation degree of the power series. Several sufficient conditions are rigorously derived as the selection principles. Finally, extensive simulation results and comparisons, with respect to both single and multirobot cases, validate that a formally computed and therefore more accurate probability of future topology can help improve the performance of active planning under uncertainty.

High-accuracy positivity-preserving numerical method for Keller-Segel model.

The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed.

Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

AbstractIn this paper, we propose a new explicit series expansion formula for the price of an arithmetic Asian option under the Black–Scholes model and Merton's jump‐diffusion model. The method is based on an equivalence in law relation together with the diffusion operator integral method proposed by Heath and Platen. The method yields explicit series expansion formula for the Asian options' prices. The theoretical convergence of the expansion to the true value is established. We also consider the American Asian option (i.e., Amerasian option) and derive the corresponding expansion formula through the early exercise premium representation. Numerical results illustrate the accuracy and efficiency of the method as compared with benchmarks in the literature.

Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps

We propose for the first time some explicit closed-form Laguerre series expansion formulas for the first passage time density functions of a jump diffusion model. Suppose that the jumps in the model are phase-type distributed. In contrast to existing methods in the literature based on numerical Laplace transform inversion, the proposed formulas are in analytical closed-form and simple to implement. Two different methods are proposed to compute the Laguerre coefficients. Various numerical examples are also given to show the effectiveness of our method.

Infinite series expansion of some finite-time dividend and ruin related functions

In this paper, we consider some finite-time dividend and ruin problems in a class of risk models under the constant dividend barrier strategy. This class of risk models includes Brownian motion risk model and the compound Poisson model with and without diffusion. By using Laguerre series expansion, we derive infinite series expansion formulas for the time T-deferred Laplace transform of the ruin time and time T-deferred expected present valued of dividend payments until ruin.

2-Variable Fubini-degenerate Apostol-type polynomials

This work deals with the mathematical inspection of a hybrid family of the degenerate polynomials of the Apostol-type. The inclusion of the derivation of few series expansion formulas, explicit representations and difference equations for this hybrid family brings a novelty to the existing literature. Moreover, certain connection formulas and several novel identities for these polynomials are established and investigated. The graphical representations of certain degenerate polynomials are explored and several new interesting pattern of the zeros are observed.

Euler and Laplace integral representations of GKZ hypergeometric functions II

We establish the intersection theory of the rapid decay homology group and formulate the twisted period relation in this setting. We claim that there is a standard method of constructing a basis of the rapid decay homology group which can be related to GKZ hypergeometric series. This can be carried out with the aid of a convergent regular triangulation $T$. When $T$ is unimodular, we can obtain a closed formula of the homology intersection number. Finally, we obtain a Laurent series expansion formula of the cohomology intersection number in terms of the combinatorics of $T$.

Euler and Laplace integral representations of GKZ hypergeometric functions I

We establish the intersection theory of the rapid decay homology group and formulate the twisted period relation in this setting. We claim that there is a standard method of constructing a basis of the rapid decay homology group which can be related to GKZ hypergeometric series. This can be carried out with the aid of a convergent regular triangulation $T$. When $T$ is unimodular, we can obtain a closed formula of the homology intersection number. Finally, we obtain a Laurent series expansion formula of the cohomology intersection number in terms of the combinatorics of $T$.

Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

Some further Hecke-type identities

In this paper, we obtain some Hecke-type identities by using two [Formula: see text]-series expansion formulae. And, the identities can also be proved directly in terms of Bailey pairs. In particular, we show that certain partial theta functions and the theta functions can be expressed in terms of Hecke-type identities.

The comparative analytical evaluation of quantum corrections to the second virial coefficient with Morse potential and its applications to real systems

In this study, the first quantum corrections to the second virial coefficient (SVC) is analytically evaluated with the Morse potential for noble gases of He,Ne,Ar,Kr,Xe and also for CH4, CO2,N2 atoms and molecules for low temperatures. This exact analytical quantum correction method includes the series expansion formula and special functions, allowing more accurate and exact evaluation of the any thermodynamic properties of matter using the second virial coefficient at wide range of temperature. To our knowledge, except for some different approximation approaches, this study is the first analytical method for the calculation of the quantum correction to second virial coefficient with Morse potential and valid for all values of parameters. The SVC quantum correction results of Morse potential are compared with the literature especially with Lennard-Jones (12-6) and indicate a good agreement at high temperatures, but the Morse potential results heavily dominate over the Lennard-Jones (12-6) potential especially for light atoms and molecules at low temperatures.

A general analytical method for evaluation of the thermodynamic properties of matters using virial coefficients with Morse potential at high temperature

AbstractIn this paper, we established an analytical formula for the second virial coefficient (SVC) with Morse potential without using any numerical methods, and the obtained formula is applied to the calculation of the speed of sound of some matter at high temperature. This approach is based on the series expansion formula and special functions, which allows the exact evaluations of any thermodynamic properties of matter using the SVC. As an application, the obtained analytical formula is used for evaluation of the SVC with Morse potential for high‐temperature gas and the plasma region of the intermolecular interactions of neutral atom gases of B, Si, Zn, H2, N2, O2, NO, CO, He, Ne, Ar, Kr, and Xe. Based on the obtained formula of SVC, the speed of sound for gases of N2, Ar, and Zn are also determined analytically. A specific maximum temperature is chosen for every gas to ensure that there are still neutral atoms in the gas, and low temperatures are avoided due to quantum effects. The results are compared with numerical data and another analytical data from the literature. The new analytical solution is shown to be in good agreement with the compared data and is verified to supply proper thermodynamic data.

Properties of Some of Two-Variable Orthogonal Polynomials

The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre–Laguerre Koornwinder polynomials and Laguerre–Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated.

Coupled consolidation and heat flow analysis of layered soils surrounding cylindrical heat sources using a precise integration technique

SummaryThe engineering applications of energy piles, geological radioactive waste disposals, and mining wells of geothermal and petroleum are usually associated with strong coupled behaviour of consolidation and heat flow. This paper aims to present an efficient precise integration technique (PIT) for the analysis of such behaviour within layered saturated soils surrounding cylindrical heat sources (ie, with a cross section as a point, ring, or disc). Each soil layer, together with its embedded part of heat source, is divided into 2N layer elements with equal thickness. Then any pair of adjacent two layer elements are merged into a heat source on the interface. With the aid of Taylor series expansion and recurrence formula for adjacent layer elements combination, such problems can be solved by means of an improved PIT. Typical examples are performed to examine the effects of heat source type and soils layered properties on the coupled consolidation and heat flow responses. The elevation of the clay surface increases with time because of thermal expansion and reaches a peak value before showing a tendency of getting stabilised because of the dissipation of pore pressure becoming dominant. Such a peak cannot be achieved in sand case because of no accumulation of pore pressure. The influencing area of the heat source was found to be limited to near the source. These quantitative results serve as good verification of the presented technique, which proves to be remarkably efficient and several orders more accurate than traditional numerical techniques in that it ideally reaches the accuracy limit of the hardware of the computers used.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Second virial coefficient of bounded repulsive potentials

Exact series expansion formulas are derived for the second virial coefficient, SVC, of a wide range of purely repulsive bounded potentials. The focus is on the consequences of the functional form, choice of disposable parameters, and temperature on the SVC characteristics. These include compressed and stretched exponential forms, and bounded inverse power (BIP) potentials, which are an extension of the inverse power system. The BIP have a quite different SVC temperature dependence to the exponential cases, characteristics which could be used to represent polymer based dispersed particles which are not accounted for by a gaussian. A new algebraic form of bounded potential is proposed whose SVC at high temperature tends to the Penetrable Sphere, but at low temperature to the BIP class.

A new approach for the design of fractional delay by an FIR filter

Fractional delay filters modeling non-integer delays are digital filters which ideally have flat group delays. In this letter, a new, simple, accurate and efficient FIR filter design to implement the digital ideal fractional delay is presented. The design technique is based on the MacLaurin series expansion formula which is applied to a given function to obtain a closed form FIR digital filter approximation of the digital ideal fractional delay operator z−α. The obtained FIR filter approximation formula of the digital ideal fractional delay operator z−α is similar to the well known Grünwald-Letnikov FIR digital filter approximation formula of the ideal analog fractional differentiator sα. Some examples have been presented to illustrate the performance and the effectiveness of this new design method. Some comparisons results show that the proposed design yields better performances than the existing designs in the literature.

Expansions of the exponential and the logarithm of power series and applications

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.