Questions tagged [laguerre-polynomials]
For questions about (associated) Laguerre polynomials, which arise in quantum physics.
27
questions
2
votes
0answers
33 views
Proving that the Laguerre polynomials do indeed solve the differential equation
I am trying to show that the Laguerre differential equation, given in my homework problem as
$xL''_n(x)+(1−x)L'_n(x)+ nL_n(x) = 0$,
is indeed solved by the Laguerre polynomials in their closed sum ...
2
votes
1answer
35 views
Sum over (squares of) Laguerre Polynomials
I'm looking for a closed form of the sum \begin{equation}
\sum_{n=0}^\infty \frac{n!}{(n+k)!} (L_n^k(x))^2 t^n,
\end{equation}
where $L_n^k(x)$ are the Laguerre Polynomials. I have been looking for ...
1
vote
0answers
25 views
A modification of the Laguerre product expansion
Given a product of Laguerre polynomials, $L_n(x) L_m(x)$, a particular question to ask is the expansion of this product in terms of the Laguerre polynomials $\{L_i(x)\}$ themselves. That is, we would ...
0
votes
0answers
17 views
Question regarding positivity for sequence of sums of integrals of Laguerrre polynomials.
Inspired by this question of a sequence of weighted sums of Laguerre polynomials with alternating signs being positive for $x>0$.
We showed that proving that the weighted sum has no zeros for $x>...
8
votes
2answers
143 views
Is the sum of the first N Laguerre polynomials (with alternating signs) always positive?
I have noticed that the following simple sum of Laguerre polynomials (weighted with alternating signs) seems to be positive for any $N$ when $x>0$:
$$\sum_{k=0}^{N}\;(-1)^{k}\;L_{k}(x)$$
More ...
0
votes
1answer
46 views
Prove $\int_0^\infty e^{-x} x^k [L^k_n (x)]^2 \, dx=\frac{(n+k)!}{n!}$
How can I prove the normalization ratio of associated Laguerre polynomials: $$\int_0^\infty e^{-x} x^k [L^k_n (x)]^2 \, dx=\frac{(n+k)!}{n!}$$ using the generator function of Laguerre polynomial, $$\...
1
vote
1answer
46 views
How to modified the Gauss–Laguerre quadrature in the case of $\int\limits_{x=0}^{+\infty}{{e^{-ax}}f(x)}dx$?
1/ How to modified the integral Gauss–Laguerre quadrature rule so that we could approximate the following integral:
$I = \int\limits_{x = 0}^{ + \infty } {{e^{ - ax}}f\left( x \right)} dx$
The things ...
4
votes
1answer
58 views
How is the Rodrigues formula $L_n^k(x)=\frac{e^x x^{-k}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+k})$ derived?
I am trying to deduce the Rodrigues formula for generalized Laguerre polynomials $$L_n^k(x)=\frac{e^x x^{-k}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+k})$$ but I have reached a point where I do not know how to ...
1
vote
0answers
36 views
Is Rodriguez's representation of the Laguerre polynomials defined $L_n^k(x)= \frac{e^xx^{-k}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+k})$?
$$L_n^k(x)= \cfrac{e^xx^{-k}}{n!}\cfrac{d^n}{dx^n}(e^{-x}x^{n+k})$$
2
votes
0answers
59 views
Find $\int\limits_0^\infty {x{e^{ - (a{x^2} + b)}}{I_0}\left( {\sqrt {c{x^2} + dx + k} } \right)dx} $?
I am trying to find the following integral:
$\int\limits_0^\infty {x{e^{ - (a{x^2} + b)}}{I_0}\left( {\sqrt {c{x^2} + dx + k} } \right)dx} $
Where, $x,a,b,c,d,k \in \mathbb{R}$ and $I_0(.)$ is the ...
2
votes
2answers
179 views
Derive Rodrigues’ formula for Laguerre polynomials
Derive Rodrigues’ formula for Laguerre polynomials
$$
L_n(x)=\frac{e^x}{n!}.\frac{d^n}{dx^n}(x^ne^{-x})
$$
The Rodrigues’ formula for Hermite polynomials can be obtained by taking $n^{th}$ order ...
1
vote
1answer
43 views
Is the $L^1$-norm of the FT of $(x+i)^n/(x-i)^{n+2}$ bounded as a sequence in $n$?
Let the function $f_n\in L^1(\mathbb R)\cap C_0(\mathbb R)$ be defined for $n\in\mathbb N$ by
$$f_n(x):=\left(\frac{x+i}{x-i}\right)^n\frac{1}{(x-i)^2}\,.$$
Then its Fourier transform is always of the ...
1
vote
1answer
77 views
Approximate a positive Schwartz function
Context: let $f \in \mathcal{S}(\mathbb{R}^+) $ be a function of the Schwartz space (all functions whose derivatives are rapidly decreasing) on $\mathbb{R}^+$.
We already know that a generic such $f$ ...
2
votes
2answers
116 views
Orthogonal polynomials with respect to $e^{-|x|} \mathrm{d} x$ on the entire real line?
The Laguerre polynomials
https://en.wikipedia.org/wiki/Laguerre_polynomials
form a system of orthogonal polynomials with respect to the measure $e^{ -x} \mathrm{d} x$ on $(0,\infty)$.
Is anything ...
2
votes
0answers
73 views
How to show orthogonality of the Laguerre polynomial $P_n(x)$?
At school, they ask me to solve this question:
For $n \in \mathbb{N}$ and $x > 0$ we define
$P_n(x) = \frac{1}{2\pi i}\int_{\Sigma}\frac{\Gamma(t-n)}{\Gamma(t+1)^2}x^t dt$
where $\Sigma$ is a ...
2
votes
1answer
47 views
How to prove the laguerre polynomial has n zero point?
Here is an equation:
$$
e^x\frac{d^n(x^ne^{-x})}{dx^n}=0
$$
Now I want to prove that this equation has $n$ different roots.
I tried to convert the equation to this form:
$$
\sum\limits_{k=0}^{n}C_n^k(-...
3
votes
1answer
51 views
Integral relation between Hermite and Laguerre polynomials
I'd like to proove the following integral relation
$$ \frac{1}{2^m m!} \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty}\,\mathrm{d}\zeta \, e^{-\zeta^2} H_m(\zeta+\zeta_1)H_m(\zeta+\zeta_2) = L_m(-2\...
1
vote
0answers
71 views
Approximating integral involving associated Laguerre polynomial
I need to numerically evaluate the following integral
$$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$
...
0
votes
0answers
37 views
Question about Laguerre polynomials formula
Is the formula equal to $$1.~L_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x})$$ or
$$2.~L_{n}(x)=e^{x}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x})\, ?$$
I'm confused with which one can I use when for ...
3
votes
2answers
99 views
Laguerre polynomial question
Can someone help me with this
$$\frac{1}{1-t}e^{-\frac{xt}{1-t}}=\sum_{n=0}^{n=\infty}L_{n}(x)\frac{t^{n}}{n!}$$
The author said that we should just expand it but I don't understand how and what $L_{n}...
1
vote
0answers
31 views
Problem with orthogonalizing the Laguerre polynomials
Alright, so I ran into a little problem while applying the Gram-Schmidt orthogonalization process. To the functions $\{1,x,x^2,x^3...\}$ over $x\in(0,\infty)$ with weight function $\sigma (x)=e^{-x}$. ...
12
votes
2answers
314 views
Prove that $(-1)^n \text{Laguerre}_n(2) \leq 1$.
I would like to prove the following inequalities on Laguerre polynomials evaluated at point 2:
$$
(-1)^n \text{Laguerre}_n(2) \leq 1
$$
This seems to hold numerically.
I tried to use the recurrence ...
1
vote
1answer
57 views
Laguerre Polynomial Termination
I had never learned much about Laguerre polynomials before, and I am trying to understand them for the first time. If we define the Laguerre equation as:
$$xy'' + (1-x)y' + \lambda y = 0$$
Then if you ...
2
votes
1answer
59 views
Recursion relation for the Laguerre polynomials
How to come to the Laguerre recursion relation ,
$$(n+1)L_{n+1}^{(\alpha)}(x)+xL_n^{(\alpha)}(x)+ (n+\alpha) L_{n-1}^{(\alpha)}(x)=(2n+1+\alpha)L_n^{(\alpha)}(x) $$
from the sum for the generalized ...
0
votes
0answers
48 views
about the Laguerre square expansion Sin(x)
The following functional series are developments using Laguerre polynomials, the second is the square of the Laguerre polynomial, the first is half as fast as the second being this is the square of ...
0
votes
0answers
15 views
Generalization of Hermite and Leguerre polynomials. Terminology and closed form expression.
Is there name for the class of polynomials of the following general form?
$P_{k,l,m}^{a}(x)=e^{-ax^{k}}\frac{d^{l}}{dx^{l}}(x^{m}e^{ax^{k}})$
The Hermite polynomials can be expressed as following:
${...
1
vote
0answers
30 views
Laguerre polynomials with integration over entire axis
The Laguerre polynomials are orthogonal with respect to the scalar product
$$
\langle f, g\rangle = \int_0^\infty f(x) g(x) \exp(-x)\,\text{d}x.
$$
Is there a class of polynomials that is orthogonal w....
