Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
649
questions
0
votes
0
answers
75
views
Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
- 31
0
votes
0
answers
83
views
Re-writing a sum of binomial coefficients as an integral of shifted Legendre polynomials
This is a question regarding the answer presented here.
In order to make this post self-contained, I am wondering if someone can explain why the sum
$$
\sum_{k=0}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{...
user1211323
0
votes
0
answers
34
views
How to construct Legendre polynomials for $x_1,...,x_k$?
I'm trying to run a nonparametric regression to estimate the unknown conditional mean $E(Y|X_1=x^*_1,X_2=x^*_2)$ using data set $\{Y_i,X_{1i},X_{2i}\}_{i=1}^n$. This could be done by nonparametric ...
- 515
2
votes
1
answer
127
views
Asymptotic equality of $\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $
Consider the Shifted Legendre Polynomial $$\tilde P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $$ where $n\in\mathbb{N}\cup\{0\}$
Question: What is the asymptotic equality of $\tilde P_n(x)$ as $n\...
- 910
0
votes
0
answers
78
views
"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$
Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
- 203
2
votes
0
answers
235
views
Integral of associated Legendre polynomials over the unit interval
I am looking for a closed-form expression for the integral of the associated Legendre polynomial $P_l^m$ over the unit interval ($l \ge m$ non-negative integers),
$$
I_l^m = \int_{0}^{1} P_l^m(x) \, ...
- 73
1
vote
1
answer
566
views
On the vanishing of the integral $\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}$ for $\ell \geq n+2$
I came across the following integral
\begin{equation}
\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}
\, ,
\end{equation}
where in the above $P_{\ell}(x)$ is the Legendre ...
user1211323
2
votes
1
answer
2k
views
First derivative of Legendre Polynomial
Legendre polynomials ($P_n$) are defined as a particular solution to the ODE.
$$(1-x^2)P_n^{''}-2xP_n^{'}+n(n+1)P_n=0$$
It is expressed by Rodrigues’ formula.
$$P_n=\frac{1}{2^nn!}\frac{d^n}{dx^n}((x^...
- 323
2
votes
1
answer
354
views
How to obtain the Legendre Polynomials from the power series solution (of the Legendre's equation)?
Solve the differential equation $$(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0.$$ Show that a polynomial, say $P_n(x)$ is a solution of the above equation, when $n$ is an integer.
I tried ...
- 3,073
-1
votes
1
answer
59
views
Legendre's Series: How can I show that $\int_{-1}^1 f(x)S(x)dx= f(1)$
Consider the series:
$$S(x)=\sum_{n=0}^{\infty}\frac{(2n+1)P_n(x)}{2}$$
Show that:
$$\int_{-1}^1 f(x)S(x)dx= f(1)$$
where $f(x)$ is any function of the interval $[−1, 1]$ on the real numbers which can ...
- 11
2
votes
2
answers
428
views
Bivariate generating function for squared binomial coefficients
I want to find a closed form to the bivariate generating function
$$
G(x, y) = \sum\limits_{i, j} \binom{i+j}{i}^2 x^i y^j.
$$
Ideally, I would prefer a direct approach that is based on the definition ...
- 1,117
1
vote
0
answers
106
views
How do I find the leading coefficient of a Legendre polynomial?
I'm trying to construct the Legendre polynomials from the differential equation. As is done in this set of lecture notes, I can get an expression for the coefficient $c_{l-2k}$ in terms of $c_l$:
$$c_{...
- 11
0
votes
1
answer
37
views
Proof of an identity of a Gaussian series over Legendre polynomials
The Legendre polynomials are a total basis (if normalized) of the real space $L^2[-1,1]$.
Let $w=w(t)=e^{-\frac{t^2}{2}}$, and $\frac{1}{2^nn!}\frac{d^n}{dt^n}[(t^2-1)^n]=P_n(t)\in L^2[-1,1]$.
$$\frac{...
3
votes
1
answer
74
views
Legendre expansion of a root polynomial
What is the Legendre expansion of $\sqrt{1-x^2}$ ? I need a closed form. Thx.
I have tried to reform it
$$
\sqrt{1-x^2}=\left. \sqrt{1-2tx+x^2} \right|_{t=x}
$$
while the generating function of ...
4
votes
1
answer
130
views
Series of product of legendre polynomials with shifted degree
I am working on some quantum mechanics and I would love to find a closed expression for the series
$$
S(x,y) = \sum_{l=0}^\infty P_l(x) P_{l+1}(y)
$$
$x,y \in \left[ -1, 1\right]$ and also for its ...
