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Results tagged with orthogonal-polynomials
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user 21907
A familly of orthogonal polynomials is a sequence of polynomials in one variable, one in each degree, such that any two of them are orthogonal with respect to some fixed scalar product on the space of polynomials. They are closely related to continued fractions and useful in harmonic analysis. There are many different families of orthogonal polynomials, among which one can cite Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.
2
votes
2
answers
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views
Zeroes of Laguerre polynomials
The simplest Laguerre polynomials are
$$
L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right).
$$
I would like to find a simple reference for proving or disproving the following assertions.
(1) All …
- 16.2k
8
votes
1
answer
530
views
Inequality for Laguerre polynomials
Let $L_n$ be the $n$-th Laguerre polynomial defined by
$\quad
L_n
(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad
$
I want to prove that
$$
\forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le …
- 16.2k
6
votes
Accepted
Multivariate Hermite Polynomials
In one dimension, you have
$$
h_k(t)e^{-π t^2}=e^{π t^2}(\frac{d}{dt})^k\bigl(e^{-2π t^2}\bigr),
$$
and $h_k$ is easily proven to be with degree $k$.
The completeness question amounts to proving
$L^2= …
- 16.2k
4
votes
Accepted
Closed formula for Hermite polynomials
Up to some normalization, the harmonic oscillator $H$ is self-adjoint such that
$$
\langle Hu, u\rangle=\sum_{k\ge 0}(\frac12+k) \vert u_k\vert^2,
$$
and thus defining a self-adjoint $A$ by the equali …
- 16.2k