# Questions tagged [hermite-polynomials]

The hermite-polynomials tag has no usage guidance.

25
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### In what sense does the Hermite expansion of a bounded smooth function converge?

Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function.
If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as
\...

0
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### Lower bound for the fractional Sobolev norm of the Hermites function

For $r \in (0, 2)$, I am interested in a lower bound for the quantity :
$$I_r(n) := \int_{\mathbb{R}} |f_n(x)|^2 |x|^{r} dx$$
where $f_n(x) = (-1)^n (\sqrt{\pi} n! 2^n)^{-1/2} e^{x^2/2} \dfrac{d^n}{dx^...

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### Closed formula for Laguerre

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...

0
votes

1
answer

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### A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$

Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial.
$$\exp\left[\...

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### Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $

Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$.
Is there a closed formula for $$\sum^{\infty}_{k=0}\...

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### Closed formula for Hermite polynomials

Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula
$$
H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .
$$
Each $H_k(x)$ is a polynomial of exact ...

2
votes

0
answers

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### Evaluation of 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-...

5
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### How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?

I got an interesting question. Consider this integral:
$$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$
It is clear that ...

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2
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### Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...

1
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1
answer

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### Integrating a B-Spline basis function with respect to the standard normal PDF

I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:
$$
\int_{-\infty}^{+\infty}B_{i}^k(u)e^{-\frac{(u-\mu)^2}{2\sigma^2}}du,
$$
where $B_i^k$ is a ...

2
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0
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### Deduce Sheffer's classification of orthogonal polynomials of A-type 0

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...

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### Complex Hermite polynomial orthogonality on weighted space

Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$
These polynomials trivially extend to functions of $w\in\mathbb{C}$...

2
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2
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### On Mehler's formula for Hermite polynomials

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the ...

1
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1
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391
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### Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) [duplicate]

I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens. ...

3
votes

0
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### Lower bound on coefficients in hermite transform of Tanh

I would like to understand $L^2\left(\mathbb{R}, \mu\right)$ approximation by polynomials of $\tanh$ (and more generally smooth functions), where $\mu$ is standard gaussian distribution. This leads to ...

2
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1
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### Hermite Transform of Tanh

I would like to understand $L^2(\mathbb{R},\mu)$ approximation by polynomials of $\tanh$ up to degree $n$ where $\mu$ is the standard Gaussian distribution. This leads to considering the Hermite ...

2
votes

1
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### A special solution to the Hermite Differential Equation

I know that the general form solution to the Hermite differential equation
$$ y''-2xy'+2\lambda y=0$$
is
$$y(x)=a_1 M(-\frac{\lambda}{2},\frac{1}{2},x^2)+a_2 H(\lambda,x),$$
where $M(\cdot,\cdot,\cdot)...

1
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0
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### Discrete Wavelet Transform and Gaussian decay

I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...

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### Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation

This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$
u_t = \frac12 u_{xx}
$$ On $\mathbb{T}$ with ...

1
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2
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### Generating function for products of complex Hermite polynomials

By making use of the generating function
$$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that
$$(*)\quad \sum_{m,n=0}^\infty \frac{u^...

4
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0
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### Turán's inequalities for Hermite functions

Given $\lambda \in \mathbb{R}$ let $H_{\lambda}(x)$ be the solution of the Hermite differential equation:
$$
\frac{d^{2}}{dx^{2}} H_{\lambda}(x)-x\frac{d}{dx}H_{\lambda}(x)+\lambda H_{\lambda}(x)=0, ...

12
votes

1
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### What are the orthogonal polynomials w.r.t. Maxwell distribution

Is there a way to get a clean presentation of the orthogonal polynomials w.r.t. the Maxwell distribution https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution ?
If you separate the ...

3
votes

1
answer

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### What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper,
http://arxiv.org/pdf/math/0401310.pdf
Has this been improved?

4
votes

1
answer

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### Asymptotic form of $L^1$-norm of Hermite functions

Background
Working on a quantum mechanics problem, I've stumbled on the problem of maximizing the functional
$$\int_{A} \varphi_m \varphi_n$$
in the limit of large $m$ and $n$, given that $n \gg m$. ...

5
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0
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### Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?

The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...