Fourier Legendre Expansion Research Articles

Legendre expansions of products of functions with applications to nonlinear partial differential equations

Given the Fourier–Legendre expansions of f and g, and mild conditions on f and g, we derive the Fourier–Legendre expansion of their product in terms of their corresponding Fourier–Legendre coefficients. We establish upper bounds on rates of convergence. We then employ these expansions to solve semi-analytically a class of nonlinear PDEs with a polynomial nonlinearity of degree 2. The obtained numerical results illustrate the efficiency and performance accuracy of this Fourier–Legendre-based solution methodology for solving a class of nonlinear PDEs.

On a higher-order version of a formula due to Ramanujan

A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1 + 1 5 ( 1 2 ) 2 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 2 + ⋯ = Γ 4 ( 1 4 ) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1 + 1 5 ( 1 2 ) 4 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 4 + ⋯ = Γ 8 ( 1 4 ) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.

A Proof of ζ(2)=π 2/6 Involving a Fourier–Legendre Expansion

We give a new proof of Euler’s famous formula using a 1954 result by M. O. González which expands the complete elliptic integral of the first kind in terms of Legendre polynomials.

Correction to: On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums

Correction to: On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums

Renormalization-group theory of the Heisenberg model in d dimensions

The classical Heisenberg model has been solved in spatial d dimensions, exactly in d=1 and by the Migdal–Kadanoff approximation in d>1, by using a Fourier–Legendre expansion. The phase transition temperatures, the energy densities, and the specific heats are calculated in arbitrary dimension d. Fisher’s exact result is recovered in d=1. The absence of an ordered phase, conventional or algebraic (in contrast to the XY model yielding an algebraically ordered phase) is recovered in d=2. A conventionally ordered phase occurs at d>2. This method opens the way to complex-system calculations with Heisenberg local degrees of freedom.

Analytical study on shear wave propagation in anisotropic dry sandy spherical layered structure

An analytical study on the propagation characteristics of shear wave in a spherical layered structure comprising of an isotropic sandy material medium covered by a concentric spherical transversely isotropic sandy material layer of finite thickness has been carried out in the present work. The source of excitation for wave propagation is considered in the outer layer and the analytical treatment with contour integration and Fourier-Legendre Expansion theorem is utilized to accomplish closed form of dispersion relation. Further, Debye Asymptotic Expansion analysis is used to deal the complication caused by the inclusion of Bessel's function, Hankel's function and their derivatives in the solution procedure. With help of Debye Asymptotic Expansion analysis, the obtained dispersion relation converts in the form of elementary functions and found to be in well-agreement with pre-established and classical results through the particular cases. Numerical computations are done to illustrate the influence of the affecting parameters (anisotropy, sandiness and radii ratio) on dispersion curves graphically. Moreover, to expose the impact of anisotropy, numerical data of three distinct transversely isotropic materials (Magnesium, Zinc and Beryl) are taken into the account to establish the comparative study with isotropic material which serves the major highlights of the present study.

A note on Clebsch–Gordan integral, Fourier–Legendre expansions and closed form for hypergeometric series

In this paper, we show that a closed-form formula for the generalized Clebsch–Gordan integral and the Fourier–Legendre expansion theory allow to evaluate hypergeometric series involving powers of the normalized central binomial coefficient \({\frac{1}{4^{n}}\genfrac(){0.0pt}0{2n}{n}}\).

Symbolic computations via Fourier–Legendre expansions and fractional operators

We investigate the interplay between the Caputo fractional operators and Fourier–Legendre expansions of hypergeometric functions, resulting in transformation formulas for series with quarter-integer parameters and in explicit evaluations of many series, including:

Chebyshev polynomial expansion of two-dimensional Landau–Fermi liquid parameters

We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau–Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier–Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermionic system differ from its three-dimensional counterpart, with the latter changing drastically. Specifically, the modified Landau–Silin kinetic equation is heavily dependent on the solution of a non-trivial contour integral specific to the 2D liquid. We find the solution to this integral and its generalizations, effectively reducing the problem of solving for the collective excitations of a collisionless two-dimensional Landau–Fermi liquid to solving for the roots of some high-degree polynomial. This analysis ultimately lays the mathematical foundation for the exploration of atypical behavior in the non-equilibrium properties of two-dimensional fermionic liquids in the context of the Landau quasiparticle paradigm.

Time-Dependent One-Dimensional Electromagnetic Wave Propagation in Inhomogeneous Media: Exact Solution in Terms of Transmutations and Neumann Series of Bessel Functions

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see [7]. In [5] using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators [8]. In [6] a Fourier–Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exact solution of the problem of the transmission of a normally incident electromagnetic time-dependent plane wave through an arbitrary inhomogeneous layer. The result can be used for efficient computation of the transmitted modulated signals. In particular, it is shown that in the classical situation of a signal represented in terms of a trigonometric Fourier series the solution of the problem can be written in the form of Neumann series of Bessel functions with exact formulas for the coefficients. The representation lends itself to numerical computation.

On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

We prove that, if the coefficients of a Fourier–Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO ( 3 ) and the principal series of the unitary representation of SU ( 1 , 1 ) .

Hypergeometry of the Parbelos

In the December 2013 issue of this Monthly, Sondow introduced a parabolic analog of the classical arbelos figure called the parbelos. He found that the ratio of the perimeter of an arbitrary parbelos to that of the corresponding arbelos is a constant S that has an elegant symbolic evaluation involving and the universal parabolic constant. Using the On-Line Encyclopedia of Integer Sequences, in 2016 Campbell experimentally discovered a hypergeometric formula for S. Zudilin then proved the formula, using recent results of Borwein, Borwein, Glasser, and Wan on the moments of Ramanujan’s generalized elliptic integrals. In the present article, we offer a variety of new proofs of Campbell’s hypergeometric formula for the parbelos constant S, including a creative proof that makes use of a Fourier–Legendre expansion.

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier–Legendre (FL) series expansions, and Fqp series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting K denote the complete elliptic integral of the first kind, for a suitable function g we evaluate integrals such as∫01K(x)g(x)dx in two different ways: (1) by expanding K as a Maclaurin series, perhaps after a transformation or a change of variable, and then integrating term-by-term; and (2) by expanding g as a shifted FL series, and then integrating term-by-term. Equating the expressions produced by these two approaches often gives us new closed-form evaluations, as in the formulas involving Catalan's constant G∑n=0∞(2nn)2Hn+14−Hn−1416n=Γ4(14)8π2−4Gπ,∑m,n≥0(2mm)2(2nn)216m+n(m+n+1)(2m+3)=7ζ(3)−4Gπ2.

On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums

In this work we continue the investigation, started in Campbell et al. (On the interplay between hypergeometric functions, complete elliptic integrals and Fourier–Legendre series expansions, arXiv:1710.03221 , 2017), about the interplay between hypergeometric functions and Fourier–Legendre ( $$\text {FL}$$ ) series expansions. In the section “Hypergeometric series related to $$\pi ,\pi ^2$$ and the lemniscate constant”, through the FL-expansion of $$[x(1-x)]^\mu $$ (with $$\mu +1\in \frac{1}{4}{\mathbb {N}}$$ ) we prove that all the hypergeometric series $$\begin{aligned}&\sum _{n\ge 0}\frac{(-1)^n(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^3,\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^4,\\&\quad \sum _{n\ge 0}\frac{(4n+1)}{p(n)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^4,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^3,\; \sum _{n\ge 0}\frac{1}{p(n)}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^2 \end{aligned}$$ return rational multiples of $$\frac{1}{\pi },\frac{1}{\pi ^2}$$ or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $$\frac{\log x}{\sqrt{x}}$$ and related functions, we show that in many cases the hypergeometric $$\phantom {}_{p+1} F_{p}(\ldots , z)$$ function evaluated at $$z=\pm 1$$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\begin{aligned} \sum _{n\ge 0}\frac{1}{(2n+1)^2}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^2,\quad \sum _{n\ge 0}\frac{1}{(2n+1)^3}\left[ \frac{1}{4^n}\left( {\begin{array}{c}2n n\end{array}}\right) \right] ^2. \end{aligned}$$ In the section “Twisted hypergeometric series” we show that the conversion of some $$\phantom {}_{p+1} F_{p}(\ldots ,\pm 1)$$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $$\sum _{n\ge 0} a_n b_n$$ where $$a_n$$ is a Stirling number of the first kind and $$\sum _{n\ge 0}b_n z^n = \phantom {}_{p+1} F_{p}(\ldots ;z)$$ .

Stress analysis of a hollow sphere compressed between two flat platens

The theoretical basis for determination of tensile strength and elastic constants by compression of hollow spheres between two flat platens is provided in this paper. The method of solution follows the displacement function approach, and three displacement functions are introduced so that the fifteen governing equations can be uncoupled analytically. The Hertz contact between the out surface of the hollow sphere and the flat loading platens is considered, and Fourier-Legendre expansion technique is applied on the Hertz contact stress in order to determine the unknown constants in the general analytical expression for stress components. An analytical solution for the stress and displacement components within a hollow sphere compressed between two flat platens is obtained. Numerical results of the analytical solution show that the stress distributions within hollow spheres are not uniform, two tensile stress concentration zones are usually developed at the inner surface and near the loading area along the loading axis of the hollow spheres within both thick and thin hollow spheres, but the maximum tensile stress is usually developed at the inner surface, except few cases for relatively thick hollow sphere with a very small Poisson ratio. The maximum tensile stress developed along the axis of loading can be used to estimate the tensile strength of hollow spheres. The curves obtained by the analytical solution for the ratio between the vertical and horizontal strains at the equator of the out surface of the hollow sphere, together with those for nominal stress versus the normalized displacement between the two flat loading platens, may provide optional way to determine Poisson's ratio and Young's modulus of the hollow sphere, respectively.

Some estimates for the error in Fourier–Legendre expansions of functions of one variable

Some issues concerning expansions of functions in Fourier–Legendre series is considered in L2[−1, 1]. In particular, the rate of their convergence in the classes of functions characterized by the generalized modulus of continuity are estimated, and estimates of the remainder terms are obtained.

On tensor product approximation of analytic functions

We prove sharp, two-sided bounds on sums of the form ∑k∈N0d∖Da(T)exp(−∑j=1dajkj), where Da(T):={k∈N0d:∑j=1dajkj≤T} and a∈R+d. These sums appear in the error analysis of tensor product approximation, interpolation and integration of d-variate analytic functions. Examples are tensor products of univariate Fourier–Legendre expansions (Beck et al., 2014) or interpolation and integration rules at Leja points (Chkifa et al., 2013), (Narayan and Jakeman, 2014), (Nobile et al., 2014). Moreover, we discuss the limit d→∞, where we prove both, algebraic and sub-exponential upper bounds. As an application we consider tensor products of Hardy spaces, where we study convergence rates of a certain truncated Taylor series, as well as of interpolation and integration using Leja points.

The analytical solutions for the stress distributions within elastic hollow spheres under the diametrical point loads

This paper presents an exact analytical solution for the stress distributions within an elastic hollow sphere subjected to diametrical point loads. The solution is suitable for both thin and thick hollow spheres. New variables are introduced in order to uncouple the system of governing equations so that explicit differential equations are obtained for displacement components and stress components. Moreover, Fourier–Legendre expansion technique is employed in order to determine the unknown coefficients in the analytical solutions for hollow spheres. The present solution can be considered as an extension of the classical solution by Hiramatsu and Oka (Int J Rock Mech Min Sci 3:89–99, 1966) for solid spheres under the point loads, which provided the theoretical basis for the point load strength test. Unlike in solid spheres, the stress concentrations within the hollow spheres under the point loads are usually developed at the joint point of the inner surface and the loading axis, and the thinner the hollow sphere, the larger the tensile stress concentrations developed at the inner surface. This numerical result indicates that the failure of the hollow spheres usually starts at the inner surface, and the normalized tensile stress at the inner surface increases with the increase in Poisson’s ratio and internal pressure, but decreases with the increase in the size of the loading area. Moreover, significant shear stress zone is usually developed in the areas immediately inside the outer surface, and the maximum shear stress is often developed at the point immediately inside the outer surface jointing the edge of the loading area and the center of the hollow sphere. The present solution can be used to analyze the failure mechanism of bulk foams made up of hollow spheres in engineering.

Differential hierarchical models for elastic prismatic shells with microtemperatures

The present paper is devoted to construction of differential hierarchical models for elastic prismatic shells of variable thickness with microtemperatures. To this end, Vekua's dimension reduction method, based on the Fourier-Legendre expansions, is applied to basic equations of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures. The special emphasis is placed on cusped prismatic shells.

Inhomogeneous strain fields within silicon spheres under the point load test and the strain effect on the quantum valence-bands

This paper presents an exact solution for inhomogeneous strain and stress distributions within silicon spheres under the point load test. The contact problem between the spherical head of the steel cones and the surfaces of the silicon spheres is considered. Displacement functions are introduced in order to uncouple the governing equations. The Fourier–Legendre expansion technique is employed so that all of the boundary conditions can be satisfied exactly. When the isotropic limit is considered, the classical solution by Hiramatsu and Oka [20] is covered as a special case. It was found that the strain distributions are relatively uniform within central part (say r/R<0.6, where r and R are the distance from the center and the radius of the sphere, respectively) of the silicon spheres under the point load test, but very high tensile strain concentrations are usually developed near r/R=0.9. In addition, based on quantum mechanics and energy band theory, the effect of strain on three quantum valence-bands of silicon is analyzed. Numerical results show that the large strain induced at r/R=0.9 of the sphere under the point load test significantly breaks some symmetric property of quantum valence bands of silicon, and has obvious effect on the constant energy surfaces and the conductivity effective masses of heavy hole (HH) band, the light hole (LH) band and the split-off (SO) band, which is closely related to optical and electric property of silicon.