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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I need help finding a generating function using some relation between the Bessel function and the Laplace integral for the Legendre Polynomials
The Bessel function of the first kind and order n has the integral representation
$J_n(z)=i^{-n}/\pi \int_0^\pi e^{iz\cos\theta}\cos(n\theta)d\theta$
By using the Laplace integral for the Legendre ...
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The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime, the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $$p-\left(\frac{-ab}{p}\right)\,.$$
I got a hint that ...
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Prove that the even (odd) degree Legendre polynomials are even > (odd) functions of $t$.
a.) Prove that the even (odd) degree Legendre polynomials are even
(odd) functions of $t$.
b.) Prove that if $p(t) = p(-t)$ is an even polynomial, then all the
odd order coefficents $c_{2j+1} ...
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Roots of Legendre Polynomial
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
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