# Questions tagged [hermite-polynomials]

200 questions

**18**

votes

**2**answers

### Hermite polynomials recurrence relation

Hermite polynomials $H_n (x)$ can be obtained using the recurrence relation
$$H_{n+1} (x)=2xH_n (x)-2nH_{n-1} (x).$$
To prove this, I started by calculating the first derivative of the Hermite's Rodrigues formula $H_n (x)=(-1)^n e^{x^2} …

Glenio Rosario

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### Solving $\left(x-c_1\frac{d}{dx}\right)^nf(x)=0$ for $f(x)$

I'm given that
$$\left(x-c_1\frac{d}{dx}\right)^nf(x) = 0$$
I have to solve for $f(x)$ in terms of $n$.
For $n=0$:
$$f(x)=0 \tag{0}$$
For $n= 1$:
$$\begin{align}
xf(x) - c_1f'(x) &= 0 \\
\quad\implies\quad f(x) &=…

Tim Crosby

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### Use of a substitution to prove that $e^{2xt-t^2}$ is the exponential generating function of the Hermite polynomials

The generating function encodes all the Hermite polynomials in one formula. It is a function of $x$ and a dummy variable $t$ of the the form:
$e^{2xt-t^2}=\sum^\infty_{n=0}\frac{H_n(x)}{n!}t^n. $
We begin by…

JamesT

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### Fourier transform of Hermite polynomial times a Gaussian

What is the Fourier transform of an (n-th order Hermite polynomial multiplied by a Gaussian)?

user153388

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### Fourier Transform of Hermite polynomial $H_n(x)e^{-\frac{x^2}{2}}$

I'm trying to solve the 2D paraxial equation $2i\partial_zu=-\partial_x^2u$, for the initial condition $u(x,z=0)=H_n(x)e^{-x^2/2}$, with $x$ and $z$ both real and $n\geq0$.
For $n=0$, I used the Fourier transform—defined as …

EarlGrey

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**7**

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### Deriving Rodrigues Formula and Generating function of Hermite Polynomial from $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$

There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, 3rd edition) she starts with the differential…

ghthorpe

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### Computing the Fourier transform of $H_k(x)e^{-x^2/2}$, where $H_k$ is the Hermite polynomial.

[Notations] The definition of Fourier transform of a $L^1$ function $f$ is given by the formula $\int f(x)e^{-ix\cdot\xi}dx$, with no normalizing factors; similarly for the Fourier-Plancherel transform on $L^2$. The Hermite polynomial of order $k$…

Dilemian

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votes

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### Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by :
$$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$
has solutions of the
$$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} $
Are there solutions to this equation for a more…

user35952

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### Weird Integral Involving Hermite Polynomials

I have stumbled upon the following integral involving the Hermite polynomials:
$$ I(m) = \int_\mathbb{R} e^{i m x} \left[ e^{-\frac{x^2}{2}} H_m(x) \right] dx \, , \quad m \in \mathbb{N} \cup \{0\} \, , $$
which is rather weird. It came up from…

QuantumBrick

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**6**

votes

**1**answer

### Hermite polynomials approximate of a function and its derivatives

Given a differentiable function $f\in \mathcal C^{(n)}(\mathbb R) \cap L^2(\mathbb R,e^{-x^2/2}dx)$ and its Hermite polynomial expansion $f_n=\sum_{i=0}^n a_i \psi_i$. Is it true that $\int_{-\infty}^\infty…

Hans

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### Inner product of scaled Hermite functions

I'm attempting to find a closed form expression for
$$\int_{-\infty}^{\infty}e^{-\frac{x^2\left(1+\lambda^2\right)}{2}}H_{n}(x)H_m(\lambda x)dx$$
where $H_n(x)$ are the physicist's hermite polynomials, but haven't had any luck. Anyone know of a way…

garserdt216

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### Asymptotic Behavior of Power Series Terms in the Hermite Equation

When solving for the wave function under a harmonic potential $V(x)=\frac{1}{2}kx^2$, we attempt a solution in the form:
$$\psi(y)=H(y)e^{-\frac{1}{2}y^2},\quad y:=\left(\frac{m\omega_o}{\hbar}\right)^{\frac{1}{2}}x,\quad \alpha:=\frac{2E}{\hbar…

Angelo Di Bella

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**5**

votes

**3**answers

### Second order inhomogeneous equation: $y''-2xy'-11y=e^{-ax}$

My question relates to an a second order inhomogeneous equation:
$$y''-2xy'-11y=e^{-ax}$$
First I need to investigate the homogeneous equation:
$$y''-2xy'-11y=0$$
$$y''-2xy'=11y$$
Forms Hermite's Equation where $\lambda = 11$
So I need a general…

Student146

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**5**

votes

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### Generation of Hermite polynomials with Gram-Schmidt procedure

I want to use the Gram-Schmidt procedure to generate the first three Hermite polynomials. Given the set of linearly independent vectors $\{1,x,x^2,...\}$ in the Hilbert space $L^2(R,e^{-x^2}dx)$, I apply the orthogonalisation procedure as…

Antonio

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### $L_2$ scalar product between Hermite polynomials

I am trying to compute the $L_2$ scalar product between (probabilists’) Hermite polynomials (defined as in Wiki) with Gaussian weight and different scales, i.e. for some constants $c, d$:
$$\frac{1}{\sqrt{2\pi}}\int…

Ester Mariucci

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