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A Fourier–Matsubara series expansion is derived for imaginary–time correlation functions that constitutes the imaginary–time generalization of the infinite Matsubara series for equal-time correlation functions. The expansion is consistent with all known exact properties of imaginary–time correlation functions and opens up new avenues for the utilization of quantum Monte Carlo simulation data. Moreover, the expansion drastically simplifies the computation of imaginary–time density–density correlation functions with the finite temperature version of the self-consistent dielectric formalism. Its existence underscores the utility of imaginary–time as a complementary domain for many-body physics. [ABSTRACT FROM AUTHOR]

Two summation theorems concerning the F 1 2 (1 / 2) -series due to Gauss and Bailey will be examined by employing the "coefficient extraction method". Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate " 1 / 2 " may serve as a reference source for readers. [ABSTRACT FROM AUTHOR]

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series. [ABSTRACT FROM AUTHOR]

On page 188 of his lost notebook, Ramanujan introduced distinct classes of exquisite infinite series, representing them in terms of Eisenstein series. The objective of this paper is to develop various differential identities related to Borwein's cubic theta functions and h-functions. Also, we present certain identities containing Eisenstein series of level 6 and h-functions that are used to establish relations involving class one infinite series and h-functions. Additionally, we present a straightforward approach to evaluate a discrete convolution sum by employing relationships involving Eisenstein series and h-functions. [ABSTRACT FROM AUTHOR]

The (p, k)-parametrization method, as introduced by Alaca, offers an innovative approach to derive a variety of new Eisenstein series identities that incorporate Borweins' theta functions. This paper presents new Eisenstein series identities that involve Borweins' cubic theta functions, achieved through the (p, k)-parameterization technique. [ABSTRACT FROM AUTHOR]

The motivation of this study is to develop effective and economical assistive technologies for people with physical disabilities. The novelty in this manuscript is the application of the average value-based technique to accurately represent the involved biomechanics of the lower limb joints during the human gait cycle. This mathematical formulation of lower limb joints' biomechanics forms the first objective for modeling and final exoskeleton prototype development. To account for modeling the characteristics of human locomotion, the nth-order linear differential equation with constant coefficients is considered with appropriate modification. The physical characteristics of an individual are represented by the constant coefficients (P 0 , P 1 , P 2 , and P 3) of the modified infinite series, which are obtained by processing experimental data collected using an optical technique. The differential terms of the infinite series are replaced by difference terms (δ b avg , δ 2 b avg , and δ 3 b avg) since the data were captured as a set of digital values. The work presented here is based on the experimental results of individuals suitably categorized according to their physical nature like age and other physical structure. The optically monitored positional values of the lower limb joints of the individual subjects while they are completing the gait cycles are used for getting values of different terms of the model. The data collected through the conduct of experiments are used for finding the values of the terms of the differential equation. The model is effectively validated through experimental results. It was determined that the representation's accuracy fell within the ± 5% acceptable tolerance limit. The model is prepared for healthy as well as disabled persons, through which the disability is quantified. The resulting model can be used to develop assistive devices for people with physical disabilities. This results in the rehabilitation process and thereby helps the reintegration into society, subsequently allowing them to lead a normal life. [ABSTRACT FROM AUTHOR]

This paper deals with the three-dimensional analysis of stress distribution in a long circular cylinder imbedded in a poroelatic medium. The surface of the cylinder is subjected to a known pressure. The equations of the classical theory of elasticity are solved in terms of an infinite series. [ABSTRACT FROM AUTHOR]

By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun ['Series with summands involving harmonic numbers', Preprint, 2023, arXiv:2210.07238v7]. This is realised by 'the coefficient extraction method' implemented by Mathematica commands. [ABSTRACT FROM AUTHOR]

Several researches have confirmed the possibility of localizing small anomalies via Kirchhoff migration (KM); however, when the background information is unknown, small anomalies cannot be satisfactorily retrieved. This fact can be examined through the simulation results; however, related theoretical result to explain the reason of such phenomenon has not yet been investigated. In this contribution, we show that the imaging function of the KM can be expressed by an infinite series of the Bessel function of the first kind, material properties, and antenna arrangement, and applied alternative value of the background wavenumber. Based on the theoretical result, we explain why the exact location and shape of anomalies cannot be retrieved. The simulation results with synthetic data exhibited to support the theoretical result. [ABSTRACT FROM AUTHOR]

In a recent paper [Cáceres MO, Comments on wave-like propagation with binary disorder. J. Stat. Phys. 2021;182(36):doi.org/10.1007/s10955-021-02699-0.], the evolution of a wave-like front perturbed by space-correlated disorder was studied. In addition, the generic solution of the field mean-value was presented as a series expansion in Terwiel's cumulants operators. This infinite series cuts due to the algebra of naked Terwiel's cumulants when these cumulants are associated to a space exponential-correlated symmetric binary disorder. We apply an equivalent approach to study the dispersion-relation for 1D surface gravity waves propagating on an irregular floor. The theory is based on the study of the mean-value of plane-wave-like Fourier modes for the propagation and damping of surface waves on a random bottom. [ABSTRACT FROM AUTHOR]

Recently, a new dispersion index, as a measures of information, has been introduced and called varentropy. In this article, we introduce new measures of variability based on two measures of uncertainty, namely, the Kerridge inaccuracy measure and the Kullback-Leibler divergence. Their generating functions are considered and their infinite series representations are given. These new measures and associated properties, bounds and illustrative examples are all presented in detail. Finally, an application of Kullback-Leibler divergence and its dispersion index is illustrated by using the mean-variance rule. [ABSTRACT FROM AUTHOR]

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems. [ABSTRACT FROM AUTHOR]

We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence. [ABSTRACT FROM AUTHOR]

We study lattice paths by combinatorial methods on the positive lattice. We give some identity that produces the functional equations and generating functions to counting the lattice paths on or below the main diagonal. Also, we consider the subdiagonal lattice paths in relation to lower triangular arrays. This presents a Riordan array in conjunction with the columns of the matrix of the coefficients of certain formal power series by implying an infinite lower triangular matrix . We derive new combinatorial interpretations in terms of restricted lattice paths for some Riordan arrays. [ABSTRACT FROM AUTHOR]

Eight infinite series involving harmonic-like numbers are coherently and systematically reviewed. They are evaluated in closed form exclusively by integration together with calculus and complex analysis. In particular, a mysterious series W is introduced and shown to be expressible in terms of the trilogarithm function. Several remarkable integral values and difficult infinite series identities are shown as consequences. [ABSTRACT FROM AUTHOR]

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data. [ABSTRACT FROM AUTHOR]

We consider the corotational Maxwell model which is perhaps the simplest constitutive model with a nontrivial oscillatory shear response that can be solved analytically. The exact solution takes the form of an infinite series. Due to exponential convergence, accurate analytical approximations to the exact solution can be obtained by truncating the series after a modest number ( ≈ 10–20) of terms. We compare the speed and accuracy of this truncated analytical solution (AS) with a fast numerical method called harmonic balance (HB). HB represents the periodic steady-state solution using a Fourier series ansatz. Due to the linearity of the constitutive model, HB leads to a tridiagonal linear system of equations in the Fourier coefficients that can be solved very efficiently. Surprisingly, we find that the convergence properties of HB are superior to AS. In terms of computational cost, HB is about 200 times cheaper than AS. Thus, the answer to the question posed in the title is affirmative. [ABSTRACT FROM AUTHOR]

The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory , but in comparison it is motivated by Bolzano's concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some differ, such as the size of rational numbers. However, set sizes are only partially, not linearly, ordered. Quid pro quo. [ABSTRACT FROM AUTHOR]

The Bessel function of the first kind J N k x is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I N k x . The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for J N k x useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving   1 F 2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k. [ABSTRACT FROM AUTHOR]

A simple yet efficient solution to the problem of transverse electric (TE) scattering by a two-dimensional right trapezoidal groove placed in a perfect electric conductor (PEC) surface is presented. Using superposition of plane waves and applying the Neumann boundary condition at the groove walls, the transverse magnetic field is expressed as the sums of infinite series of waveguide modes. Considering the equivalent magnetic current on the aperture, a Fredlholm's integral equation with logarithmic singular kernel is extracted and then discretised by a collocation method based on a piecewise constant approximant. The extracted linear system of equations can be easily solved by the matrix methods for the unknown coefficients. Numerous examples are presented to verify the results and show the ability of the proposed method. Finally, this method is employed to examine the effect of the groove parameters and incidence angle on the scattering signatures. [ABSTRACT FROM AUTHOR]

Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results. [ABSTRACT FROM AUTHOR]

This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI). [ABSTRACT FROM AUTHOR]

This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index κ ∈ (0 , 2 ] . We provide a new test statistic and show that its asymptotic distribution under the null hypothesis of non-stationary I (1) series is a functional of a stable process. When the change point in persistence is not known, the consistency is always given under the alternative, either from stationary I (0) to non-stationary I (1) or vice versa. The proposed tests can be used to identify the direction of change and do not over-reject against constant I (0) series, even in relatively small samples. Furthermore, we also consider the change point estimator which is consistent and the asymptotic behavior of the test statistic in the case of near-integrated time series. A block bootstrap method is suggested to determine critical values because the null asymptotic distribution contains the unknown tail index, which results in critical values depending on it. The simulation study demonstrates that the block bootstrap-based test is robust against change in persistence for heavy-tailed series with infinite variance. Finally, we apply our methods to the two series of the US inflation rate and USD/CNY exchange rate, and find significant evidence for persistence changes, respectively, from I (0) to I (1) and from I (1) to I (0) . [ABSTRACT FROM AUTHOR]

The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We construct two new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the three-dimensional complex projective space. In the first series the sheaves have an even first Chern class, and in the second series they have an odd one, while the second and third Chern classes can be expressed as polynomials of a special form in three integer variables. We prove the uniqueness of components in these series for the Chern classes given by those polynomials. [ABSTRACT FROM AUTHOR]

The inhomogeneous heat equation for a semi-infinite cylindrical solid body with mixed boundary conditions of the first and second kinds on the surface of the cylinder was solved using the Laplace and Hankel integral transforms. The solution of the mixed boundary-value problem was reduced to the dual integral equations which were solved using an appropriate substitution, and their solution was finally reduced to the Fredholm integral equations of the second kind which were solved by the infinite series method. [ABSTRACT FROM AUTHOR]

In this paper, a theorem dealing with the |N, pn|k summability factors of infinite series has been generalized to |A, pn, β; γ|k summability method by using (Φ, δ)- monotone sequences. This new theorem also includes some new results. [ABSTRACT FROM AUTHOR]

This article is in response to a reference to my work as a work in which the expression for the density of the charge induced by the point charge between two infinite plane-parallel plates is in the form of an infinite series. Indeed, in my article there were formulas that I obtained for the Green's function, the surface density of the induced charge on plane-parallel plates written as infinite series. However, the article was devoted to the method of conformal transformation of the initial problem to a simpler one, which makes it possible to obtain analytical expressions for the mentioned quantities. In the present article, I would like to clarify the situation and outline the advantage of the conformal transformation method taking into account the axial symmetry of the problem as compared to solving the problem in the form of infinite series. [ABSTRACT FROM AUTHOR]

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation's assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. [ABSTRACT FROM AUTHOR]

In this work, perforated and restrained nanobeams with deformable boundary conditions are modeled based on the non-local strain gradient elasticity theory with the elastic medium effect. In this approach in which the Fourier infinite series and Stokes' transformation are used together, the nanobeam is detached into parts from the two boundary points with the main part. Then using elastic force boundary conditions, a system of linear equations in terms of nonlinearity, elastic medium parameter, spring coefficients and small size effects are derived and the eigenvalue solution of these equations is also presented. The nonlinear stability of restrained nanobeam is examined and the effects of size parameters, perforation and elastic spring coefficients are studied. To reveal the accuracy and effectiveness of the offered model, several numerical applications are solved for the nonlinear stability response of restrained nanobeams with elastic medium effects. The outcomes of this method validated that the presented approach is appropriate for the stability behavior of rigid and restrained nanobeams with perforated cross section. [ABSTRACT FROM AUTHOR]

This paper presents new analytical expressions for displacements Green's functions to a steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source. These results are obtained on the base of special structural formulas for displacements Green's functions, which are expressed in terms of respective Green's functions for Poisson's equation. An example of the application of derived new analytical expressions is presented for a particular spatial BVP for a thermoelastic unbounded parallelepiped, subjected to a constant heat source, given inside of a rectangle. Both analytical expressions for displacements Green's functions and thermoelastic displacements in the case of a particular problem are obtained in the form of double infinite series, containing product between exponential and trigonometric functions, which satisfy basic equations, boundary conditions on the marginal strips and vanishes at infinity. The presented example of a new steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source will permit readers to derive the other examples to new analytical expressions for Green's functions. These Green's functions can be applied as kernels in the method of the boundary integral equations to solution of many particular BVP for thermoelastic unbounded parallelepiped. All these analytical results can be used also as some test problems for different numerical methods. [ABSTRACT FROM AUTHOR]

Few new Eisenstein series identities involving Borwein’s theta functions are formulated using (p, k)- parametrization technique which is introduced by Alaca. As an application, we have evaluated the convolution sum of divisor functions which are related to Eisenstein series. [ABSTRACT FROM AUTHOR]

In 2011, Guillera ['A new Ramanujan-like series for $1/\pi ^2$ ', Ramanujan J. 26 (2011), 369–374] introduced a remarkable rational ${}_{7}F_{6}(\frac {27}{64})$ -series for ${1}/{\pi ^2}$ using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall's ${}_{5}F_{4}$ -sum. Another proof of Guillera's ${}_{7}F_{6}(\frac {27}{64})$ -series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall's sum together with the WZ method. Subsequently, Chen and Chu introduced a q -analogue of Guillera's ${}_{7}F_{6}(\frac {27}{64})$ -series. The many past research articles concerning Guillera's ${}_{7}F_{6}(\frac {27}{64})$ -series for ${1}/{\pi ^2}$ naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational ${}_{7}F_{6}(\frac {27}{64})$ - and ${}_{6}F_{5}(\frac {27}{64})$ -series for $\sqrt {2}$. [ABSTRACT FROM AUTHOR]

In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on ℝ∞ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given. [ABSTRACT FROM AUTHOR]

Simple and accurate closed-form approximations to the probability density function of the sum of independent identically distributed random variables are derived. We propose a unified approach capable of providing the pdf of the sum when the summands involve the generalized Nakagami-m, generalized Gamma, Rayleigh and Weibull random variables. Unlike others work, the approximations do not use any infinite series or require any complicated special functions. Numerical results show that the new approximations have satisfactory accuracies. As an application, the outage probability and the M-QAM average SER performance of pre-detection EGC diversity are presented. [ABSTRACT FROM AUTHOR]

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerategammafunction, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums. [ABSTRACT FROM AUTHOR]

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann-Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau-Ginzburg-Higgs equation and the Sine-Gordon one. Some numerical simulations are plotted and a conclusion is given at the end. [ABSTRACT FROM AUTHOR]

By applying the "coefficient extraction method" to hypergeometric series, we establish several remarkable infinite series identities about harmonic numbers and five binomial coefficients, including three conjectured by Z.-W. Sun. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the equivalence of two matrix summability methods. Some known results are also presented. [ABSTRACT FROM AUTHOR]

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Kratzer potential. We do the same for a radial power-law potential with inverse-square and inverse-cube singularities. These solutions are written as infinite convergent series of Bessel functions with a discrete index. As physical application of the latter solution, we treat electron scattering off a neutral molecule with electric dipole and electric quadrupole moments. [ABSTRACT FROM AUTHOR]

By computing definite integrals, several infinite triple series are explicitly evaluated in terms of and as well as Catalan's constant . [ABSTRACT FROM AUTHOR]

The present article considers an anti‐plane stress problem of three cracks at different orthotropic materials' interfaces. According to the geometry of the problem, the governing equations and mixed boundary conditions have been formulated. Fourier integral transformation is used to convert the mixed boundary value problem into dual integral equations, which gives two equations containing infinite series. The investigation of the problem concerning anti‐plane cracks subjected to static loadings is done with the help of the Schmidt method to satisfy the given boundary conditions. The difference in displacements is expanded to proceed further in the problem, which becomes zero outside the cracks. Numerical computations are carried out for the graphical representation of stress intensity factors (SIFs) at all cracks' tips. The Interaction among the cracks as those are in close proximity to each other or move away are represented pictorially. Detailed numerical results and discussion are done for the considered materials, which include aluminium, epoxy and graphite epoxy. The novelty of the present article is the numerical analysis and pictorial presentation of SIFs at the tips of interfacial offset parallel cracks for various crack lengths and normalised heights for different combinations of materials. The authors have obtained variations in SIFs for the cracks at the interfaces of dissimilar composite materials. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties related to a multi-index special function W α ¯ , ν ¯ that arose from an eigenvalue problem for a multi-order fractional hyper-Bessel operator involving Caputo fractional derivatives. We show that for particular values of the parameters involved in this special function W α ¯ , ν ¯ , this leads to the hyper-Bessel function of Delerue. The Laplace transform of the W α ¯ , ν ¯ is discussed obtaining, in particular cases, the well-known functional relation between hyper-Bessel function and multi-index Mittag-Leffler function, or, quite simply, between classical Wright and Mittag-Leffler functions. Moreover, it is shown that this multi-index special function satisfies a recurrence relation involving fractional derivatives. In a particular case, we derive, to the best of our knowledge, a new differential recurrence relation for the Mittag-Leffler function. We also provide derivatives of the 3-parameters function W α , β , ν with respect to parameters, leading to infinite power series with coefficients being quotients of digamma and gamma functions. [ABSTRACT FROM AUTHOR]

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers Hn(p,q) ... in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values. [ABSTRACT FROM AUTHOR]

There are many ways to approximate π, such as the Leibniz formula or Machin-like formula. However, in this classroom capsule, we provide a series for π using the Maclaurin series of arcsine, using a sequence defined by sine. Also, we explain why the approximation gets better via defining a double sequence and approximating error via Taylor's remainder theorem. [ABSTRACT FROM AUTHOR]

Thermoelastic dissipation or thermoelastic damping (TED) can restrict the quality factor of micro/nanoring resonators seriously. This paper employs the non-Fourier model of Guyer–Krumhansl (GK model) to render a size-dependent formulation and analytical solution for approximating the amount of TED in micro/nanorings with circular cross-section by inclusion of nonlocal and single-phase-lagging effects. To fulfill this objective, the equation of heat conduction in the ring is first established according to GK model. Then, by placing the temperature distribution obtained from the heat conduction equation in the TED relation defined on the basis of thermal energy approach, an expression in the form of infinite series is given for TED, which includes non-classical parameters of GK model. Finally, after checking the validity of the model through a comparative study, several simulation results are prepared to emphasize on the influence of different factors such as non-classical parameters of GK model, geometry of ring, vibrational mode and ambient temperature on TED value. Numerical examples reveal that the mentioned factors along with the two- or three-dimensional heat transfer (2D or 3D) model have major influences on TED variations. [ABSTRACT FROM AUTHOR]