# Questions tagged [hermite-polynomials]

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### Polynomial mean [closed]

I am not able to justify the above inequality, contained in Garcia-Cuerva, Rubio De Francia "Weighted Norm Inequalities and Related Topics", pag.292-293, at the end of 292. Let $P_N(f)$ be ...
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### Polynomial Interpolation $p(0)=1$ and $p'(-1)=p(1)$

The following practice problem was given: Find all polynomials $p(x)$ that satisfy the interpolation conditions: $p(0)=1$ $p'(-1)=p(1)$ and the degree of $p(x)$ as small as possible. I don't know ...
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### Do Hermite polynomials satisfy $\int(H_n(x))^2 e^{-x^2} dx= 2n \int (H_{n-1}(x))^2 e^{-x^2} dx$?

I'm trying to prove that the norm of the Hermite polynomials (physicist's version) equals $2^n n!$. I stumbled upon this answer and I don't understand parts of the proof. First of all, it seems ...
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### How to calculate Hermite interpolation without slope in the end points (unknown derivatives)?

In class, the following exercise was proposed. Determine the natural cubic spline that passes through the points (0,1) (1, 0) (2, 0.5) (3, 1). Use the Hermite method I need to interpolate a function, ...
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### How to find weight and points of half interval Gauss Hermite quadrature?

I want to find weight $w_i$ and points $x_i$ such that $$\int_0^\infty \exp (-x^2)f(x)dx \approx \sum_{i=1}^n w_if(x_i)$$ Is there useful reference for this? I'm implementing half interval Gauss-...
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### Formulas's deduction from the generating function of Hermite polynomials

In the book "Essential Mathematical Methods for Physicists" comes the following problem that I am trying to solve: at first I could see that the first formula that is given as an answer is ...
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### How is the formula $H_{2n+1}(x)=(-1)^n\sum_{s=0}^{n}(-1)^s(2x)^{2s+1}\cfrac{(2n+1)!}{(2s+1)!(n-s)!}$ derived?

How the relationship $H_{2n+1}(x)=(-1)^n\sum_{s=0}^{n}(-1)^s(2x)^{2s+1}\cfrac{(2n+1)!}{(2s+1)!(n-s)!}$ is deduced ,working with Hermite polynomials ? There is a problem so I don't know how to deduce ...
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### Proving $\frac{1}{e}\cosh(2x)=\sum_{n=0,1,2,…}^\infty\frac{1}{(2n)!}H_{2n}(x)$, where $H_{2n}(x)$ is the even order Hermite polynomial

How can we prove the following relation $$\frac{1}{e}\cosh(2x)=\sum_{n=0, 1, 2, ...}^\infty\frac{1}{(2n)!}H_{2n}(x)$$ where the $H_{2n}(x)$ is the even order Hermite polynomial? I didn't get any clue ...
1answer
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### Hermite polynomial generating function

How would I write the following polynomial in terms of the Hermite polynomials, $H_n(z)$? \begin{equation} P_n(z) = \sum_{k=0}^{[n/2]} \frac{n!a^k}{k!(n-2k)!}(2a z)^{(n-2k)} \end{equation} I have ...
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### integral of hermite polynomials

Objective: Show that $$\int^{\infty}_{-\infty} x e^{-x^2} H_n(x) H_m(x) dx = \pi^{1/2} 2^{n-1} n! \delta_{m,n-1} + \pi^{1/2} 2^n (n+1)! \delta_{m,n+1}$$ My attempt at this is: \begin{eqnarray*} \...
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### Fourier transform of Hermite polynomial via generating function

I've spent a lot of times trying to show that $$\mathcal{F}[e^{-x^2/2} G(x,t)] = e^{-k^2/2} G(k, -it)$$ with $G(x,t)$ being the generating function of Hermite polynomial, $$G(x,t) = e^{2tx - t^2}$$...
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### What is the difference between probabilistic and physicists Hermite polynomials?

In the equation they differ only by a fraction of 2 and in the results one is without a constant in x^n and one has those constants. Why are they defined differently and what are their separate uses?
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### Hermite - Interpolation

This is my first question here. I hope I can find an answer to my question. I tried to find the answer in Books, Videos, Scripts and german forums (I'm german). But nobody could help me. It's about ...
3answers
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