# Questions tagged [hermite-polynomials]

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### General hermite basis?

I am trying to understand the Wikipedia entry on Hermite polynomials and I am having issues. Let's say I have $N$ points and $N$ derivatives at those points. If I did not have the derivatives I could ...
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### Hermitian interpolation problem

Calculate the interpolation polyonmial $p$ of the Hermitian interpolation problem (i) to the data $$x_0=0, x_1=1, y_0^{(0)}=1, y_1^{(0)}=2, y_0^{(1)}=1, y_1^{(1)}=2$$ and check the derivative values. (...
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### Solving second order ODE - Hermite polynomial and Kummer confluent hypergeometric function

I have the following system of two second order ODEs: \begin{align} (L -D)V+Q^{-1}g(x)=0 \end{align} where \begin{align} L = \frac{1}{2} \sigma_{x}^{2} I \partial_{x}^{2}+(A - a_1 x) I \partial_{x} \...
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### Integral connection with Hermite and Legendre polynomials

Show that $$\int\limits_{-\infty}^{+\infty}x^n e^{-x^2} H_n(tx) dx =\sqrt{\pi} n! P_n(t)$$ Case seems rather complex, I'm completely stuck...
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### "Easy" uniform bound for hermites functions.

The hermites functions are defined as $\psi_n(x) = (-1)^n(2^n n! \sqrt{n})^{-1/2} e^{x^2/2} \frac{d}{dx^n} e^{-x^2}$. They satisfies many properties (see https://en.wikipedia.org/wiki/...
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### An expression involving Hermite polynomials at imaginary values

Let $H_n(x)$ be the $n$-th (probabilists') Hermite polynomial. I need to bound from above the expression: $$\frac{1}{d}\sum_{n=0}^{d-1}\exp(-t^2/2)\frac{\vert H_n(it)\vert^2}{n!}$$ where $t>0$ is ...
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