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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.

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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 …
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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= …
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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 …
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2 votes
2 answers
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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 …
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