关于Legendre函数的一致渐近展开

The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion

We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.

Sampling theorem based Fourier-Legendre transform

The product of any number of Legendre functions, under a restricted domain, can be expanded by the corresponding Legendre polynomials, with the coefficient being the sinc function. While an analogous expansion can be made for any number of Gengenbauer functions, it is not allowed for more than two Jacobi functions. To obtain such an expansion, the sampling theorem is of great availability.

Graph rules for the linked cluster expansion of the Legendre effective action

Graph rules for the linked cluster expansion of the Legendre effective action $Γ[ϕ]$ are derived and proven in $D\geq 2$ Euclidean dimensions. A key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs. The hopping interaction is allowed to be long ranged and scale dependent, thereby producing an in principle exact solution of $Γ[ϕ]$'s functional renormalization group equation.

Asymptotic Bessel-function expansions for Legendre and Jacobi functions

We present new asymptotic series for the Legendre and Jacobi functions of the first and second kinds in terms of Bessel functions with appropriate arguments. The results are useful in the context of scattering problems, improve on known limiting results, and allow the calculation of corrections to the leading Bessel-function approximations for these functions. Our derivations of these series are based on Barnes-type representations of the Legendre, Jacobi, and Bessel functions; our method appears to be new. We use the results, finally, to obtain asymptotic Bessel function expansions for the ro…

Generating Functions for Lacunary Legendre and Legendre-like Polynomials

The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new properties regarding the generating functions, repeated derivatives of irrational functions and so on. We finally extend the method of the integral transform to functions with more than one variable and derive new expansion criteria of a given function …

两种谱方法在一类均匀内热源对流问题中的应用

Application of two spectral methods to a problem of convection with uniform internal heat source

本文采用两种基于Fourier级数展开的方法（一种基于Fourier级数展开的方法和一种基于移位Legendre多项式展开的方法），对具有均匀内热源的水平流体层中的线性对流问题的特征值问题进行了解析研究。对每种方法都作了一些理论上的评述。给出了数值结果，并与已有的结果进行了比较.良好的协议被发现。