Tagged Questions
2
votes
1answer
50 views
Integral identity with square of Jacobi polynomial
This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$:
...
1
vote
1answer
35 views
Irreducibility of the Gegenbauer polynomials
The Gegenbauer polynomials $C_n^{(\alpha)}(x)$ can be defined by requiring that they satisfy that
$$ \frac{1}{(1-2xt+t^2)^{\alpha}} = \sum_{n=0}^{\infty} C_n^{(\alpha)}(x)t^n.$$
In the cases when ...
0
votes
1answer
32 views
Using a known recursion relation to solve an Integral
Provided that the following generating expression defines the Hermite polynomials $ H_n(x)$
$$
F(x,h)= e^{2hx-h^{2}} = \sum_{n=0}^{\infty} H_n(x) \frac{h^{n}}{n!}
$$
Find a recursion relation and ...
1
vote
0answers
35 views
Connection between Hermite & Legendre polynomials
Prove that $$H_n(x)= 2^{n+1}e^{x^2}\int_x^\infty e^{-t^2}t^{n+1}P_n\left(\frac{x}t\right)dt,$$
where $H_n$ is Hermite polynomial & $P_n$ is Legendre polynomial
1
vote
1answer
57 views
Proving property of Legendre Polynomial (by induction?)
Let $P_n(x) = \dfrac{1}{2^n n!}\dfrac{d^n}{dx^n} [(x^2-1)^n]$, we know that $\sum\limits_{n=0}^{\infty} P_n(x)t^n=(1-2tx+t^2)^{-1/2}$
How can we proved
$P_{2n}(0)=\dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ ...
1
vote
2answers
56 views
Orthogonality of Legendre Functions
The Legendre Polynomials satisfy the following orthogonality condition:
The definite integral of $P(n,x) \cdot P(m,x)$ from $-1$ to $1$ equals $0$, if $m$ is not equal to $n$: $$\int_{-1}^1 P(n,x) ...
3
votes
2answers
186 views
Calculate an integral involving Hermite polynomials
I have to calculate the integral
$$\frac{1}{\sqrt{2^nn!}\sqrt{2^ll!}}\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}H_n(x)e^{-x^2+kx}H_l(x)\;\mathrm{d}x$$
where $H_n(x)$ is the $n^{th}$ Hermite ...
2
votes
2answers
297 views
Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$
What is the value of $\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}H_n(x)dx$ where $H_n(x)$ is the $n^{\small\mbox{th}}$ Hermite Polynomial (physicist's convention)?
0
votes
1answer
57 views
Zernike and Legendre polynomials
The even and odd Zernike polynomials are defined as follows:
$$Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$$
and:
$$Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$$
with:
...
0
votes
0answers
45 views
if system is localized
Consider function $g=e^{-|x|}, x \in R$.
Let $\psi_n$ be a Hermite function (see http://en.wikipedia.org/wiki/Hermite_polynomials for definition). Consider system $\Psi=\{\psi_n\}, n \in N$.
Let ...
1
vote
2answers
478 views
Problems regarding integrals involving Legendre polynomials
I am finding difficulty doing this integral involving Legendre polynomials.
$$\int_{-1}^1 x^2 P_{n-1}(x)P_{n+1}(x)dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}$$ I have two strategies in my mind both of ...
1
vote
1answer
355 views
Proving a property of Legendre polynomials containing its derivatives
I am trying to prove the following property of Legendre polynomials.
$$nP_n(x)=x{P_n^\prime(x)} - P^\prime_{n-1}(x)$$
My guess is that I somehow have to use the Bonnets recursion formula
...
3
votes
1answer
157 views
Chebyshev polynomial properties [duplicate]
Possible Duplicate:
Chebyshev polynomial question
I am trying to prove a property of Chebyshev polynomials.
Given the polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined ...
2
votes
1answer
83 views
Proving an identity involving the derivative of the Laguerre polynomials with respect to $n$
I've recently come across the following equality in a paper: suppose one defines an analytic function $L(n,x)$ which is equal to the $n$th Laguerre polynomial for $n\in\{0,1,\ldots\}$, and let* ...
3
votes
2answers
85 views
A question of the norm calculation of Hermite function.
Define the Hermite function $H_n (x)$ by $$H_n (x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ then prove that $$ \int_{\mathbb R} |H_n (x) |^2 e^{-x^2} dx = 2^n n! \sqrt{\pi}$$
1
vote
0answers
81 views
What is the correct differential equation for the Laguerre function?
I would like to derive the correct Laguerre function from the differential equation
but the differential equations seems different from the original one.
What is the correct differential equation and ...
0
votes
1answer
150 views
Seeking for some neat function for Hermite polynomial
Let us define $$F_n=\int f(z) |He_n(z)|^2 \, dz \, dz^*$$ is there any type of function $f$ could make that $F_n=0$ for $n\geq 2$ and $F_n>0$ for $n<2?$
...
1
vote
1answer
170 views
Operator for Laguerre polynomial
Is there any operator that could truncate Laguerre polynomial so that the polynomial is only left with the highest order term?
2
votes
0answers
120 views
Common zeros of associated Legendre functions
Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
2
votes
2answers
293 views
Why are the Gegenbauer polynomials called “ultraspherical”?
There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre ...
1
vote
0answers
359 views
How to prove that Legendre polynomials form a complete basis using functional analysis
I would like to prove that the Legendre polynomials form a complete basis on the interval [-1, 1] using functional analysis.
Here is what I came up with so far. Legendre polynomials $P_n(x)$ are ...
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
5
votes
1answer
609 views
Connection between Legendre polynomial and Bessel function
In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation
$$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$
valid for $x>0$.
...
4
votes
2answers
222 views
Finding some orthogonality in a convolution-like integral over Legendre polynomials
I encountered the following integral in my (physics) research, and I've yet to find an analytic solution:
$$I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) ...
2
votes
1answer
191 views
Solving Shallow water Equations with Hermite polynomials
I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line.
The main equation is
$$\frac{d^2\eta}{dt^2} + ...
7
votes
2answers
488 views
The roots of Hermite polynomials are all real?
The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$
How does one prove that all the roots of the Hermite polynomial are real?
1
vote
1answer
461 views
Associated Legendre polynomials of fractional order
My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$, i.e $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are both ...
1
vote
0answers
166 views
On the completeness of the generalized Laguerre polynomials
I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$.
1. Orthonormality
\begin{equation}
\int_0^{\infty} ...
4
votes
1answer
190 views
Orthogonality of the Gegenbauer Polynomials
Typically the orthogonality relation for the Gegenbauer polynomials is given as:
$$
...
2
votes
2answers
313 views
The relationship between Legendre Polynomials and monomial basis polynomials
I am currently doing filter designs and stumbled across this mathematical problem which I cannot understand. I was hoping for some insight from experts around this field to help me with this.
...
1
vote
3answers
300 views
(Should be easy) Legendre polynomial integration debugging
I am trying to evaluate the following integral:
$$I_n=\int\limits_{-1}^1 f(x)P_n(x)dx$$ where $f(x)=1$ for $x\in[-1,0)$ and $f(x)=-1$ for $x\in(0,1]$ and $P_n(x)$ is the Legendre polynomial of degree ...
8
votes
1answer
120 views
Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$
In answering a question on math.SE, I attempted to find integral of Fejér kernel by using
$$
K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right)
$$
where $U_n(z)$ stands for the ...
3
votes
1answer
481 views
Differentiation of generating function of Hermite's polynomials
The generating function of Hermite's polynomials is given by $G(x,t)=e^{2xt-t^2}$ for $x, t \in \mathbf{R}$. It is known that $\displaystyle G(x,t)=\sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}$ for $x, t ...
4
votes
2answers
624 views
Integrating Legendre Polynomials over half range
Solving for the potential of a conducting sphere with hemispheres at opposite potentials, (not using Green's function) I am stuck at this point:
$$I_l = V_1 \int_0^1 P_l(x)dx+V_2 \int_{-1}^0 ...
0
votes
2answers
308 views
Proving a Laguerre polynomial integral
After a fair bit of effort, I managed to prove that
$$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$
where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with ...
5
votes
1answer
541 views
Integral Representations of Hermite Polynomial?
One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
14
votes
5answers
2k views
Roots of Legendre Polynomial
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
...
