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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4 votes
2 answers
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How to solve this Hermite Initial Value Problem with Fourier Transform

So I have a Hermite-Gaussian profile given by the initial value problem $$ iu_z + u_{xx}=0,\quad (x,z)\in\mathbb{R}\times\mathbb{R}_+ $$ where the condition initially is given by this $$ u_0(x) = H_n\...
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1 answer
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Expectation Hermite polynomials product $H_n(X) H_k(Y)$ when $X,Y$ are dependent Gaussians

If $X,Y\sim N(0,1)$ with $E XY=\rho$ covariance, what is the expectation of their product after applying probabilist's Hermite polynomials $E_{X,Y} H_n(X) H_k(Y)$? My initial guess is $\rho^{2k}\...
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Hermite interpolation formula [closed]

Now a very trivial question about this book Ralston: A first course in numerical analysis the Hermite formula example, how it follows $$\frac{-1}{32768}<E(0.60)<\frac{-1}{2^{23}}$$ from this: $...
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Expression for $ \left( \frac{d^2}{dx^2} + \frac{c}{x} \frac{d}{dx} \right)^k \phi(x)$

Consider the following differential operator \begin{align} D_c= \frac{d^2}{dx^2} + \frac{c}{x} \frac{d}{dx} \end{align} defined on $x>0$ for some given non-zero constant $c$. Let $\phi(x)=\exp(-x^...
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1 answer
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How do I continue the scheme to find the Hermite polynomial?

Please help me comlete the scheme for finding the Hermite polynomial. I was given: f(o)=2, f´(0)=1, p(1)=3, p(2)=10. I solved the problem as follows , but at the end I get two 2s, whish give 0 in ...
1 vote
0 answers
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Closed form for infinite sum over Hermite polynomial $H_{2n}$ [closed]

I am looking for a closed form of the following infinite sum $$ \sum_{n=0}^{\infty} \frac{\left(n-\frac{1}{2}\right)!\ (r)_n }{(2n)!} w^{2n}H_{2n}(z)$$ I found a similar formula on Wolfram as $$ \sum_{...
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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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What exactly does it mean for the Hermite polynomials to be a bases for $L^2(\mathbb{R})$ functions?

In a lot of source it is mentioned that the Hermite polynomials are an orthonormal bases (with respect to a specific inner product) of $L^2(\mathbb{R})$ functions. Does this mean that for any $f \in L^...
2 votes
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Generalization of Hermite polynomials for negative, fractional degree.

After solving a differential equation I've gotten a solution given as a linear combination of Hermite polynomials and confluent hypergeometric functions. The caveat is that the Hermite polynomials ...
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1 answer
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How do I write a power series from a recurrence relation? [closed]

I have ODE $$y''- 2xy'+ 2ky = 0$$ From this I have found a recurrence relation $$C_{n+2} = \dfrac {(2n+2k)}{(n+1)(n+2)}C_n$$ How would I write this out in series summation form? I am trying to relate ...
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Sequence of largest roots of Hermite polynomials

Let's denote with $\{x_n\}_{n\in N}$ the sequence of the largest root of the (statistical) Hermite polynomial $h_n$. Much is known about upper and lower bounds of the $x_n$, see for example here. ...
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Obtain a Sequence of Orthogonal Polynomials from their orthogonality relation.

I have been asked to show explicitly a Sequence of Orthogonal Polynomials that satisfy the following orthogonality relation \begin{equation} \displaystyle\int_{-\infty}^{\infty} H_m(x)H_n(x)e^{-x^...
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Is there a connection between Brownian motion and Hermite polynomials?

Given a standard one dimensional Brownian motion (starting in 0) $(B_t)_{t \geq 0}$ I am supposed to proof that $B_t$, $B_t^2 - t$ and $B_t^3 - 3tB_t$ are martingales, which I was able to do without a ...
1 vote
1 answer
42 views

Integral of a squared Hermite Polynomial with a Gaussian that is not mean 0

I am interested in integrals of the following form: $$\text{I}(\mu,\sigma^2) = \int_{-\infty}^\infty dxH_n^2(x)e^{-(x-\mu)^2/2\sigma^2} $$ I know that in the case when $\mu = 0$ and $\sigma^2 = 1/2$ ...
1 vote
1 answer
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Interpretation of the expansion form of a Hermite Polynomial

The two forms of the Hermite polynomial $$H_{n}(y)=\left( 2y -\frac{d}{dy} \right)^n\cdot 1$$ $$H_{n}(y)=(-1)^n e^{y^{2}}\frac{d^n}{dy^n}e^{-y^{2}}$$ can be proven to be equivalent by induction (e.g ...
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Find Hermitian interpolation with Lagrangian polynomials

Hi I have an exercise that I cannot solve. Can someone help me? I have to find the solution to the following Hermitian interpolation problem p(x) under the conditions: $p(x_0)=-1, \ p'(x_0)=1 \ p(x_1)=...
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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 ...
7 votes
3 answers
184 views

Evaluating $\int_0^\infty (t+a)^k e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\mathrm dt$, $k\in\Bbb N_0$

Let $$ I_k=\int_0^\infty (t+a)^k e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\mathrm dt, $$ with $k\in\Bbb N_0$ and $a>0$. Since $k$ is an integer we can expand the binomial to obtain $$ ...
1 vote
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$\overline{\text{span}(h_n(x))} = L^2(\mathbf{R}) \iff$ if $f \in L^2(\mathbf{R})$ satisfies $(f,h_n)=0$ for all $n$, then $f=0$?

In the context of Hermite Polynomials (here) it is stated that $\overline{\text{span}(h_n(x))} = L^2(\mathbf{R}) \iff$ if $f \in L^2(\mathbf{R})$ satisfies $(f,h_n)=0$ for all $n$, then $f=0$. How ...
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Suggestions to evaluate a summation involving Hermite polynomials

I started with the the following four-variable function, $f(s,x,y,u)$, expressed as a summation involving the product of Hermite polynomials. $f(s,x,y,u)=\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{(n-...
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1 answer
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A closed-form for higher-order derivatives of the Dawson integral?

We have as the Dawson integral $$ \mathcal D(x):=e^{-x^2}\int_0^xe^{t^2}\,\mathrm dt. $$ I am interested in an expression for $\mathcal D^{(n)}(x)$ that does not involve sums. For example, we could ...
2 votes
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Summation involving product of hypergeometric functions

I started with the following sums involving the product of Hermite polynomials. $f(x,y;u)=\sum_{m=0}^{\infty}\frac{H_{2m}(x) H_{2m}(y)}{(2m)!}\big(\frac{u}{2}\big)^{2m} \tag{1}$ $g(x,y;u)=\sum_{m=0}^...
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Comparing the different bases for representing a function

Suppose I have some function $f(x)$. I know that this function can be represented in several different bases. For example, we can express this using Legendre polynomials, Hermite polynomials, Fourier ...
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Evaluation of integrals involving Hermite polynomials

For $p,q=1,3,5,\cdots $, I would like to evaluate certain integrals involving the Hermite polynomials. Recall that the $n$-th Hermite polynomial is defined as $$H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-...
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Evaluation of a summation involving the use of Mehler's formula

I came across an infinite summation of the following form involving the product of two Hermite polynomials while solving a physics problem. $f(x,y;u,a,b)=\sum_{n=0}^{\infty} \frac{(a+bn)H_{n}(x)H_{n}(...
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Gaussian integral of Hermite polynomials

I am looking for a simple expression (in terms of sums of polynomials for instance) of the integral $$I = \int_{-\infty}^{+ \infty} dx \int_{-\infty}^{+ \infty} dy \, H_{k}(x+y) H_{l}(x-y) \, e^{-2x^...
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Probabilistic Hermite Polynomials weight function and interval of integration

With the weight function $w(x)=\exp^{-\frac{x^2}{2}}$ and the interval of integration $[-\infty, \infty]$ one can obtain Probabilistic Hermite polynomials: $H_0(x)=1$, $H_1(x)=x$,$H_2(x)=x^2-1$, $H_3(...
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Hermite interpolation - I have an answer, but don't know why does it actualy work.

I was asked to use Hermite Interpolation to find an interpolating polynomial of a degree at most $5$ that satisfies the conditions: $x = 0$ $x = 1$ $ x= 3$ $f(x) = $ $1$ $3$ $1$ $f'(x) = $ $0$ $3$ ...
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Find a fundamental solution for the operator $-\Delta + {|x|}^2$ in $\mathbb{R}^n$ using Hermite expansions.

For clarity, $\Delta := \sum_{j=1}^n \frac{\partial^2}{\partial x_j^2}$ denotes the $n$-dimensional Laplace operator. This is Problem 27 in Exercise 8 of Strichartz's A Guide to Distribution Theory ...
1 vote
1 answer
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Matrix exponential of infinite antisymmetric matrix with entries only next to its diagonal

What is the exponential $\exp (t A)$ of the operator $A$ whose components are given by $A_{nm} = \delta_{nm-1} \sqrt{n+1} - \delta_{nm+1}\sqrt{n}$ where the $n,m \in \mathbb{N}_0$. If we just consider ...
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Integral of a single Hermite polynomial

Is there a solution to this definite integral $$ \int_{-\infty}^{d*} \left(\sqrt{c_0\; x^2 +c_1}\right)^{-m} H_{-m}\left(c_2\; \sqrt{\frac{c_0\; x^2}{c_0\; x^2 + c_1}}\right) \mathrm{d} x $$ with $...
2 votes
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Infinite sum of normalized Hermite polynomials

Is it known any closed-form function of the infinite sum of the normalized probabilist's Hermite polynomials, i.e., \begin{align} \sum_{j=0}^\infty \frac{He_j(t)}{\sqrt{j!}} \end{align} where $He_j(t)...
2 votes
1 answer
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Hermite polynomials for non-integer degree

I have solved an eigenvalue problem using Mathematica and the answer is in terms of Hermite polynomials. Now, for integer degrees $H_n(z)$, I can find a nice definition. However, in the solution to ...
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Hermite Curve to Approximate Sin(X) X and Y values incorrect

So I'm trying to approximate sin(x) using Hermite curves. I have four separate curves that when graphed are supposed to look similar to sin(x) on the period 0 through 2PI. I am doing the calculations ...
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How to prove that $p$ is an Hermite polynomial?

Problem: Let $$H_n(x):=(-1)^n e ^{\frac{x^2}{2}} \frac{d^n}{dx^n}(e ^{-\frac{x^2}{2}})$$ be the $n^{th}$-Hermite polynomial and $$G_n(t,x):=t^{\frac n 2}H_n(\frac {x}{\sqrt{t}})$$ Moreover, let $(B_t)...
1 vote
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Approximate orthogonality between two sets of Hermite functions.

Consider the set of Hermite functions $\{\phi_{n}(x,\varepsilon_{1})\}_{n}:= A$ defined below. \begin{equation} \label{eqn:funcs} \phi_{n}(x,\varepsilon_{1}) = \frac{\sqrt[8]{1+\big(\frac{2\...
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Uniform interpolation on a Cubic Hermite Spline

I have a 3D spline with points $p_0,p_1,...,p_n$ and tangents $m_0,m_1,...,m_n$. I'm using the formula described in this page. $p(t) = (2t^3-3t^2+1)p_0+(t^3-2t^2+t)m_0+(-2t^3+3t^2)p_1+(t^3-t^2)m_1$, ...
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Why $\frac{(-i)^n}{\sqrt{2\pi}} \int_{-\infty}^\infty \sqrt{x^2+p^2} e^{ipx} H_n(p) e^{-p^2/2} dp \approx \sqrt{2n+1} H_n(x) e^{-x^2/2}$?

Why the below holds for $n\geq 1$? $$\frac{(-i)^n}{\sqrt{2\pi}} \int_{-\infty}^\infty \sqrt{x^2+p^2} e^{ipx} H_n(p) e^{-p^2/2} dp \approx \sqrt{2n+1} H_n(x) e^{-x^2/2}.$$ The Hermite polynomial is ...
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Computing the Fourier Transform of $\exp(-\log(x)^2)$

I would like to use the approach of R. E. Fredericksen and R. F. Hess, “Estimating multiple temporal mechanisms in human vision,” Vision Res., vol. 38, no. 7, pp. 1023–1040, Apr. 1998, doi:10.1016/...
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1 answer
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Prove for general form of function at -x containing derivatives of order n

I have stumbled across multiple casses of functions (explicitly Hermit and Legendre polynomials) for which I wanted to prove the symmetry. While doing so I always ended up with the following equations:...
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44 views

Is it possible to find the value of a polynomial from the generating function?

Suppose I've to find $H_4(0),$ where $H$ represents the Hermite polynomial. I've only been provided with the following relation : $$e^{-t^2+2tx}=\sum_n H_n(x)\frac{t^n}{n!}$$ My first step is to ...
1 vote
1 answer
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Gram Schmidt process to generalize the orthogonal sequence $(-1)^ne^{x^2/2} \frac{d^n}{dx^n}(e^{-x^2})$

Let us consider a sequence of functions $\{f_n(x)\}_{0}^{\infty}$ given by $$f_n(x)=x^{n}e^{-x^2/2},~~~~\forall~ x \in \mathbb R,~~n \geq 0.$$ Then the sequence is basically, $~\{e^{-x^2/2},~xe^{-x^2/...
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Hermite polynomials for fractional orders and negative arguments

Hermite polynomials of order $n\in\mathbb{N}_0$ can be expressed as a special case of the confluent hypergeometric function (also called Tricomi's confluent hypergeometric function): $$H_n(x) = 2^n U\...
5 votes
2 answers
223 views

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)^...
1 vote
1 answer
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Orthogonal polynomial associated with $Lf= -f'' -f' + x^2f$ =0?

Consider the following equation $Lf= -f'' -f' + x^2f$ =0$ Can we construct orthogonal polynomials from the solution of this differential operator? The general shape for classical orthogonal polynomial ...
2 votes
1 answer
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Relationship between Hermite coefficients of $g(x) := f(\alpha x)$ and those of $f$

For a function $f: \mathbb R \to \mathbb R$ which is square-integrable w.r.t the $e^{-x^2}dx$, its $n$th Hermite coefficient is defined by $$ c_n(f) := \int_{-\infty}^\infty e^{-x^2}H_n(x)f(x)dx, $$ ...
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