Questions tagged [laguerre-polynomials]

For questions about (associated) Laguerre polynomials, which arise in quantum physics.

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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
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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 ...
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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 ...
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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>...
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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 ...
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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
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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
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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 ...
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0answers
36 views

Is Rodriguez's representation of the Laguerre polynomials defined $L_n^k(x)= \frac{e^xx^{-k}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+k})$?

$$L_n^k(x)= \cfrac{e^xx^{-k}}{n!}\cfrac{d^n}{dx^n}(e^{-x}x^{n+k})$$
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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
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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
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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
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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
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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
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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
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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
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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\...
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0answers
71 views

Approximating integral involving associated Laguerre polynomial

I need to numerically evaluate the following integral $$\sqrt{\frac{n!(n+1)!}{(n+\alpha)!(n+1+\alpha)!}}\int_0^\infty \frac{1}{\sqrt{x+c}}x^\alpha e^{-x}L_n^\alpha(x)L_{n+1}^\alpha(x)\;\mathrm dx$$ ...
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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
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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}...
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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
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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 ...
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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
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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 ...
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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 ...
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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: ${...
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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....