# Questions tagged [legendre-polynomials]

For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.

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### Approximating $\sin(x)$ on $[−1, 1]$ in $L^2$ with a second degree polynomial

$p_n : [−1, 1] → \mathbb{R} (n \in \mathbb{N})$ is a polynomial of degree $n$. $p_n$ is an orthonormal system, $\int_{-1}^{1}p_n(x)p_m(x)dx=\delta_{m,n}$. The first task was to calculate $p_0, p_1$ ...
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### Legendre expansion of the Dirac delta function

There is a known expansion for the Dirac delta function in terms of the Legendre polynomials as $\delta(x) = \sum_{k = 0}^{\infty} (-1)^k \frac{(4k + 1) (2k)!}{2^{2k + 1} (k!)^2} P_{2k}(x)$. I would ...
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### Associated Legendre functions recurrence formula

I have generating function for Associated Legendre functions : $$g(z,t)=(2m-1)!!\frac{(1-z^2)^{m/2}t^m}{(1-2zt+t^2)^{m+1/2}}$$ I need to find generalization of Bonnet's recursion formula (I need to ...
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### Integrating Legendre polynomials

I need to solve following integral: $I_{n}=\int_{-1}^{1}\frac{1}{x}P_{n}(x)P_{n-1}(x)dx$. I have hint that following equation needs to be used: $(n+1)I_{n+1}+nI_{n}=2$. Does anyone have idea how to ...
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### Show that $P_n(x) ={}_2F_1\left(-n,n+1;1;\frac{1-x}{2}\right)$.

I am told that $$P_n(x) ={}_2F_1\left(-n,n+1;1;\tfrac{1-x}{2}\right),$$ where $P_n(x)$ is Legendre polynomial and ${}_2F_1\left(a,b;c;z\right)$ is hypergeometric function. I am just wondering how to ...
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