Abstract In this present paper we apply the Lie group theory associated fractional calculus to obtain the symmetries of the $$\zeta (t)$$ ζ ( t ) -KdV fractional partial differential equation, which is given in terms of the $$\zeta (t)$$ ζ ( t ) -Riemann-Liouville time fractional partial derivative, in which a particular case of the $$\zeta (t)$$ ζ ( t ) -Hilfer fractional partial derivative is obtained. The calculus of symmetries consider the explicit formula of this infinitesimal extension of the fractional operator. The fractional partial equation is reduced to a fractional differential ordinary equation, and an analytical solution is proposed in the form of a power series. We obtain a nonlinear recurrence for the coefficients of the series. We discuss the linearized case for the fractional KdV equation, obtaining the Mainardi function as the solution, and the results are interpreted graphically. We then present the conservation law theorem for fractional $$\zeta (t)$$ ζ ( t ) operators, and we applied the law to find the law associated with each symmetry.

The Johnson system of distributions provides a flexible framework for modeling continuous random variables through monotonic transformations of the standard normal distribution. The system is typically presented in transformation form, which obscures the analytic structure of its quantile functions. This paper introduces a novel formulation: explicit power-series expansions for the Johnson SU, SL, and SB quantile functions, expressed in terms of a centered standard normal quantile function. These Johnson Power-Series Expansions (JPSE) provide a new analytic foundation for the Johnson system and facilitate structured quantile-based modeling. The SU and SB expansions consist of odd powers, but only SU has strictly positive coefficients, ensuring monotonicity for all truncation orders. In contrast, the SB expansion involves alternating-sign coefficients and may violate monotonicity under truncation. The SL expansion includes both even and odd powers with positive coefficients, requiring careful selection of truncation order to preserve validity. We derive convergence rates, remainder bounds, and sufficient conditions for monotonicity, and demonstrate through Q–Q and density comparisons that low-order truncations yield accurate approximations across the Johnson families. These results support the development of quantile-parameterized distribution systems for applications in expert elicitation and empirical quantile fitting.

This article presents the Restraining Stiffness Method (RSM), an analytical approach for buckling analysis of open-section members under general loading. Based on Bleich’s coefficient of restraint method, RSM adopts power series, introduces a new restraining body concept, and allows transverse translation along junctions. It is validated against analytical (Bulson, Bleich) and numerical (FEM, finite strip) methods, showing a general error of about 2%. RSM requires fewer elements, converges quickly, and can identify the buckling-critical plate. Currently applicable to consecutive right-angled plates, RSM overcomes traditional analytical methods’ limitations and offers an efficient, accurate tool for complex structural analysis.

Dirichlet's series associated with some power series

Growth of power series with nonnegative coefficients, and moments of power series distributions

About the algebraic closure of formal power series in several variables

This paper presents an advanced six-step linear multistep method of order five, based on second derivative non-hybrid block backward differentiation formula (BDF), developed for numerical solution of stiff systems of ordinary differential equations. The development of the multistep collocation approach is carried out using the matrix inversion techniques. The block structure facilitates simultaneous computation of multiple solution points, improving computational efficiency. Moreover, the power series is adopted as basis function for driving the discrete and continuous formulations. The analysis of the method such as consistency, zero-stability, order and error constants is presented, confirming its suitability for stiff systems. Numerical experiments on standard are tested on stiff and non-stiff ordinary differential equations showed the new method outperforms existing method in terms of accuracy.

In Mathematics, biology, physics and engineering, nonlinear Volterra integral equations (NVIEs) of the first kind are frequently encountered when modelling dynamic systems. However, because of their ill-posed nature and nonlinear terms, they present considerable difficulties. This work presents a hybrid methodology that combines a power series expansion with the Upadhyaya transform, a flexible tool from the Laplace family, building on recent developments in integral transforms. This combination resolves nonlinearities through systematic coefficient matching in the series domain and simplifies the handling of convolution kernels via the transform. We describe the fundamentals of the approach, show how it can be applied to four benchmark problems taken from earlier research, and expand it to a new case involving trigonometric nonlinearity. With an emphasis on computational clarity and verification, each example is broken down step-by-step. The results show that the hybrid approach outperforms standalone methods in terms of flexibility and ease, producing exact solutions when feasible and convergent approximations otherwise. There is potential for this method to be applied more widely in solving integral models in the real world.

This research presents the establishment of numerical solutions for space fractional Schrodinger problem (SFSP) by utilizing significant collocation methods, called fractional Clique collocation method (FCCM). First of all, the SFSP is reduced into a system of ordinary differential and algebraic equations by means of fractional Clique polynomials with collocation points. Secondly, the resulting system is solved numerically by Residual power series method (RPSM).Therefore, FCCM is a combination of fractional Clique polynomials and RPSM. Finally, illustrative examples are given to present how FCCM is striking and appealing.

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark pair production in association with a photon or a jet at the Large Hadron Collider (LHC), evaluated through \mathcal{O}(\epsilon^2) 𝒪 ( ϵ 2 ) in the dimensional regularisation parameter, \epsilon ϵ . These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation. The helicity amplitudes are expressed as linear combinations of algebraically independent components of the \epsilon ϵ -expanded master integrals–known as pentagon function–with the corresponding rational coefficients written in terms of momentum-twistor variables. We derive differential equations for the pentagon functions and solve them numerically using the generalised power series expansion method.

In this paper, we propose a new three-point iterative scheme for solving nonlinear equations, which achieves seventh-order convergence. The method begins with a standard Newton iteration, followed by two weighted-Newton steps constructed using power series expansions.The present manuscript enhances the order of convergence by integrating divided difference techniques with power series approaches, leading to an efficient and reliable iterative process.The order of convergence has been established rigorously as seven, and the corresponding error equations are derived to validate the theoretical results. A comprehensive convergence analysis is carried out, encompassing both local and semilocal convergence aspects. The local convergence results are obtained under assumptions involving only the first derivative of the operator, and a computable radius of convergence is derived. Moreover, the uniqueness of the solution within this radius is also discussed in detail. For the semilocal analysis, we employ the majorizing sequence technique, which ensures convergence from a wider range of initial approximations. Extensive numerical experiments are performed to demonstrate the validity and accuracy of the proposed method. The calculated results show excellent agreement with the theoretical predictions, confirming the robustness and efficiency of the new algorithm, particularly when compared in terms of the number of iterations and the approximated computational order of convergence.

Due to the limitations of the Laplace transform in solving certain classes of nonlinear partial differential equations, this study introduces a novel numerical technique that combines the Laplace transform with the residual power series approach to effectively handle the nonlinear time-fractional Whitham–Broer–Kaup (WBK) equations. The newly developed method, called the Laplace-Residual Power Series Method (L-RPSM), integrates the strengths of both techniques to overcome the restrictions encountered when each method is applied individually. In this approach, the fractional derivative is defined in the sense of the Caputo derivative. To demonstrate the applicability and efficiency of the proposed method, the nonlinear time-fractional WBK equations are solved step by step, and the obtained approximate solutions are analyzed in detail. The performance of the L-RPSM is validated through a numerical application, where the results are presented in tables, two-dimensional plots, and three-dimensional graphs. The results show that the L-RPSM is not only robust and efficient but also straightforward to implement. It provides rapidly convergent series solutions with very high accuracy, confirming its capability to approximate the exact solutions of the WBK equations. In addition, the flexibility of the method allows its application to various fractional partial differential equations, confirming its role as a reliable and versatile approach for analyzing nonlinear fractional systems.

The second part of the article discusses the accuracy of generalized asymptotic series representing the stress and displacement fields associated with the tip of an acute crack in orthotropic materials in the formulation of the plane problem of the theory of anisotropic elasticity. We compare the exact analytical solution to the problem of stretching an infinite anisotropic plane with an inclined central crack obtained using methods of the complex variable theory and an approximate solution found using the power series expansion method. For the first time, the fields of absolute error allowed for the truncation of the asymptotic series on a different number of terms for materials with a cubic symmetry of elastic properties are obtained. The analysis of the fields of the decadic logarithm of absolute errors showed that near the crack tip of all types of combined (mixed) deformation there are geometric points – loci of accuracy, in which the approximate solution almost coincides with the exact solution, which can be used in the interpretation of experimental and computational data obtained in order to determine the coefficients of asymptotic series for stresses and displacements near the tip of a crack or notch. To quantify the errors allowed when truncating the asymptotic series on the th term, a relatively accurate analytical solution obtained on the basis of the theory of the function of a complex variable the -norm is introduced into scrutiny, which gives an opportunity to choose and specify the number of terms of the series necessary to achieve the required accuracy when presenting the asymptotic ansatz with a multi-coefficient asymptotic series, for a wide range of slope angle values of the crack to the vertical axis (the axis of action of the applied tensile load) and the angle, setting the location of the anisotropy axes of the elastic properties of the material.

The Voigt function is the convolution of a Gaussian and a Lorentzian. We rederive power series for its half width at half maximum for the limiting cases of near-Gaussian and near-Lorentzian line shapes. We thereby provide independent verification and slight corrections of the expansion coefficients reported by Wang et al (2022). Results are used in our implementation of function voigt_hwhm in the open-source library libcerf.

Diffusion on curved surfaces deviates from the flat case due to geometrical corrections in the evolution of its moments, such as the geodesic mean square displacement. Moreover, anomalous diffusion is widely used to model transport in disordered, confined, or crowded environments and can be described by a temporal subordination scheme, leading to a time-fractional diffusion equation. In this work, we analyze the dynamics of time subordinated anomalous diffusion on curved surfaces. By using a generalized Taylor expansion with fractional derivatives in the Caputo sense, we express the moments as a temporal power series and show that the anomalous exponent couples with curvature terms, leading to a competition between geometric and anomalous effects. This coupling indicates a mechanism through which curvature modulates anomalous transport.

We propose an algorithm for expanding a formal triple power series into a three-dimensional corresponding to this series associated continued fraction, which generalized one of the simplest algorithms for expanding a formal power series into a corresponding continued fraction, namely the Viskovatov algorithm.

Let $f$ be an entire functions $f\colon \mathbb{C}^{p}\to\mathbb{C}$, represented by power series of the form$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z), z\in\mathbb{C}^p$$where $P_0(z)\equiv a_{0}\in\mathbb{C}$, $P_k(z)=\sum\limits_{\|n\|=\lambda_k} a_{n}z^{n}$ is a homogeneous polynomial of degree $\lambda_k\in\mathbb{N}$ and $ 0=\lambda_0< \lambda_k\uparrow +\infty$ ($1\leq k\uparrow +\infty$).In this paper, we present conditions such that the equality $$\rho_f:=\varlimsup\limits_{r\to +\infty}\frac{\ln\ln M_f(r)}{\ln r}=\rho(f,\mathbf{K}):=\varlimsup\limits_{r\to +\infty}\frac{\ln\ln M_f(r,\mathbf{K})}{\ln r}$$holds, where $$M_f(r):=\sup\{|f(z)|\colon |z|\leq r\},\quad M_f(r,\mathbf{K}):=\sup\{|f(z)|\colon z\in \mathbf{K},|z|\leq r\}$$and $\mathbf{K}$ is a real cone in $\mathbb{C}^p$ such that the set $\mathbb{C}\overline{\mathbf{K}}$ is non-pluripolar in $\mathbb{C}^p$.

The paper considers the problem of analytical continuation of special functions by branched continued fractions. These representations play an important role in approximating of special functions that arise in various applied problems. By improving the methods of studying the convergence of branched continued fractions, several domains of analytical continuation of the special function $H_4(\alpha,\delta+1;\gamma,\delta;-\mathbf{z})/H_4(\alpha,\delta+2;\gamma,\delta+1;-\mathbf{z})$ in the case of real and complex parameters are established. To prove the analytical continuation, the so-called PC method is used, which is based on the principle of correspondence between a formal double power series and a branched continued fraction. An example is provided at the end.

This paper examines a replacement model in which system replacement is carried out either at a random time [Formula: see text] or upon failure, rather than at a fixed time. The lifetime model is formulated for parallel systems with a random number of components, where each component follows a power series distribution. The optimization problem is investigated under a preventive age-replacement policy. The planned replacement time is assumed to be a random variable with a distribution from the proportional hazard rate family, and the lifetimes of system components are modeled using a Weibull distribution. The number of components is assumed to follow geometric, logarithmic, or zero-truncated Poisson distributions. The total expected cost rate function is used as the optimization criterion. Both analytical and numerical results are obtained, and the applicability of the proposed approach is demonstrated using a real data set.

Let k be an algebraically closed field of characteristic zero, and let k\llbracket z\rrbracket be the ring of formal power series over k . We provide several characterizations of right amenable finitely generated subsemigroups of z^{2}k\llbracket z\rrbracket , where the semigroup operation \circ is composition. In particular, we show that a subsemigroup S=\langle Q_{1},Q_{2},\ldots, Q_{k}\rangle of z^{2}k\llbracket z\rrbracket is right amenable if and only if there exists an invertible element \beta of zk\llbracket z\rrbracket such that \beta^{-1}\circ Q_{i} \circ \beta =\omega_{i} z^{d_i} , 1\leq i \leq k , for some integers d_{i} , 1\leq i \leq k , and roots of unity \omega_{i} , 1\leq i \leq k .