Questions tagged [legendre-polynomials]

Question 1

From the following formula for the Legendre polynomials $P_n(x)$: $$P_n(x) = \frac{1}{2^n}\sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \binom{n}{k}\binom{2n-2k}{n} x^{n-2k}\tag{*}$$ we can see that when $x\...

Question 2

Legendre Equation is $$(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0$$ two series solutions to Legendre Equation is$$\begin{aligned} y_1(x)&=1-\frac{\alpha(\alpha+1)}{2!}x^2+\frac{\alpha(\alpha-2)(\alpha+1)(...

Question 3

Is there a way to evaluate the following infinite sum $$ \sum_{n=1}^\infty n(n+1)(n+2) \, \xi^{n-1} P_{n+2}(\xi) \, , $$ where $P_n(\xi)$ denotes the Legendre polynomial. Here, $\xi \in [0,1]$. I ...

Question 4

Define the rising factorials $(a)_n$. The squared central binomial coefficient is $$ \frac{\left(\frac12\right)_n^2}{(1)_n^2}=\frac{1}{2^{4n}}\binom{2n}{n}^2. $$ A well-known result is given by ...

Question 5

Define shifted Legendre polynomials with Gauss hypergeometric representation, $$ P_n(2x-1)=(-1)^n\,_2F_1(-n,n+1;1;x). $$ It can be shown that, for $f\in L(0,1)^2$, such that $$ f(x)\sim\sum_{n=0}^{\...

Question 6

Consider the function $f(x)=\sum_J a_J P_J(x)$ where $x\in[-1,1]$, where $P_J(x)$ were Legendre polynomials and $a_J$ the fixed value. First part of the question Given that $f(x)=0$ for $x\in(-1,1)$, ...

Question 7

The Golub-Welsch algorithm is a technique to find the roots of a family of orthogonal polynomials numerically. According to the paper, any family of orthogonal polynomials has a three term recurrence: ...

Question 8

While working on my research, I ended up finding an infinite sum of the kind: \begin{equation} S = \sum_{l=2}^{\infty} \frac{2l+1}{l^2 (l+1)^2} P_l (\cos(\theta)) \end{equation} with $P_l$ the ...

Question 9

I was trying to understand (at a really high level) how Gauss–Legendre quadrature works and, from what I understand, it allows to exactly compute the result of an integral for a given polynomial of ...

Question 10

Heyhey, recently I came across the following problem. I want to calculate the integral over the product of a spherical harmonic and the derivative of another (not necessarily the same) spherical ...

Question 11

For $l \in \mathbb{N}$, let $P_l(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n$ be the $l$-th Legendre polynomial. Moreover, let $C(\psi) = \{x \in \mathbb{S}^2|~ x \cdot e_3 \geq \cos(\psi)\}$ be ...

Question 12

I have a function that is defined as: $$C(\theta)=\sum_{l=2} \frac{2l+1}{2l(l+1)}D_lP_l(\cos(\theta))$$ and I have this integral: $$S_{1/2}=\int_{-1}^{1/2}C(\theta)^2 \,\mathrm d(\cos(\theta))$$ ...

Question 13

I am reading Analysis I by Herbert Amann and Joachim Escher and I am trying to do Exercise IV.1.12, which states the following. I have no other tools that the basic rules for derivatives such as ...

Question 14

Define complete elliptic integral of the first kind by $K\left ( \sqrt{x} \right ) =\frac{\pi}{2} \,_2F_1\left ( \frac12,\frac12;1;x \right ) =\int_{0}^{\frac\pi2} \frac{\text{d}\theta}{\sqrt{1-x\...

Question 15

I was wondering if there is a general solution to this integral: $$\int_0^\pi P_j(\cos\theta) P_k(\cos\theta) \, \text{d}\theta $$ Or, re-written in terms of $ x = \cos\theta $: $$\int_{-1}^1 \frac{1}{...

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Questions tagged [legendre-polynomials]

Why are the values of $P_n(2m+1)$ integers?

How to show that the general formula for Legendre polynomial $P_n(x)$

Evaluating $\sum_{n=1}^\infty n(n+1)(n+2) \, \xi^{n-1} P_{n+2}(\xi),$ for $\xi \in [0,1]$

Expansions with squared central binomial coefficient ${\binom{2n}{n}^2}$ and connections to Fourier-Legendre expansions with harmonic numbers

Mellin Inversions of Terminate Multiple Hypergeometric Summations

Is it possible for the Legendre polynomial to sample a step funciton of finite height and then a second step function?

Golub-Welsch algorithm for legendre polynomial seems wrong? [closed]

Closed-form of an infinite Legendre polynomial sum

Points constraining integral of polynomial

What is the expansion of the derivative of a spherical harmonic in spherical harmonics?

Decay rate of the difference of two Legendre polynomials

Integration using Legendre polynomials give different result than Simpson integration

Roots of Legendre's Polynomials

Polylogarithm representations of $\int_{0}^{1} \frac{K\left ( \sqrt{x} \right ) }{\sqrt{x} } \operatorname{Li}_2(x)\text{d}x$ and others

General solution for integrating unweighted products of Legendre Polynomials [closed]

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