Questions tagged [laguerre-polynomials]
For questions about (associated) Laguerre polynomials, which arise in quantum physics.
58
questions
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Give a lower bound of $h(x)=\sqrt{\frac{x}{4n-x}}g^2(x)+\sqrt{\frac{4n-x}{x}}f^2(x)$ on $[\frac{1}{n-1},4n-4]$.
Let $n\in\mathbb{N}_{+}$ and $f(x), g(x)\in C^2[0,+\infty)$ be two functions satisfying
$$f(0)=0,g(0)=1$$
$$f^{'}(x)=-\frac{1}{2}g(x)$$
$$g^{'}(x)=\frac{4n-x}{2x}f(x).$$
Prove that
$$h(x)=\sqrt{\frac{...
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votes
1
answer
67
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How to rearrange $py'' + (2l+2-2p)y' + 2(n-l-1)y = 0$ into $2py'' + (2l+2-2p)y' + (n-l-1)y = 0$?
I have a 2nd order homogenous ODE:
$$py'' + (2l+2-2p)y' + 2(n-l-1)y = 0$$
where y is a function of p and n and l are variables
Its solution is the associated Laguerre polynomials $L_{n-l-1}^{2l+1}(2p)$...
- 105
1
vote
1
answer
41
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Find Special Function from Series Form?
I obtained this list of series (there are more but listed up to 5th order) and I suspect they are related to the Laguerre polynomials. Strictly speaking they are not $L_n^{\alpha}(x)$ but something ...
- 448
1
vote
1
answer
292
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Addition formula for generalized Laguerre polynomials
For the Hermite polynomials, there is the following addition formula
Is there a similar formula for the generalized Laguerre Polynomials, in particular for $a=b=1/2$.
I.e. what is
$L^m_k(0.5x + 0.5 y)...
- 69
1
vote
0
answers
46
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Integral involving Bessel functions, exponential and two Laguerre polynomials
in the context of a physics problem, I encountered the following integral:
$$
\int_0^{\infty} d x J_{N_1+N_2}\left(q x\right) \cdot x^{\left|N_1\right|+\left|N_2\right|} e^{-\frac{x^2}{2}} L_{a_1}^{\...
- 11
0
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1
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37
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How to evaluate directly Associated Laguerre Polynomial multiplied with $e^{-x/2} x^{k/2}$?
The Associated Laguerre Polynomial orthonormality relation reads:
\begin{align}
\int_0^{\infty} dx f^k_n(x) f^k_m(x) = \delta_{n,m}
\end{align}
Where,
$$
f^k_n(x) = \sqrt{\frac{n!}{(n+k)!}} e^{-x/2} x^...
- 469
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0
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Searching for weighted-$L^1$ summable orthonormal basis of $L^2(0,\infty)$
so I was working on something and bumped into the following question:
Given some $a>0$, does there exist a complete orthonormal system $ (f_n)_{n \in \mathbb{N}} $ of $L^2(0,\infty)$ such that $\...
- 256
3
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0
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68
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Fractional Laguerre function $L_{n-\frac{1}{2}}(x)$
Is there any formula to represent Laguerre functions with fractional index (in this case only divided by 2) in terms of Bessel functions $I_0(x)$ and $I_1(x)$?
I found this formula in Wolfram ...
- 3,164
1
vote
1
answer
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Alternative roots of generalized Laguerre polynomials
$\require{\physics}$
Hi,
I am wondering if it is possible to approximate the roots of the generalized Laguerre polynomial $L_n^{(\alpha)}(x)$ not with respect to $x$ but with respect to $n$, i.e. ...
- 55
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votes
0
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How to modified the Gauss–Laguerre quadrature in the case of $\int\limits_{10}^{\infty}{{e^{-x}}f(x)}\mathrm{d}x$?
I am not sure if we change the lower limits, How to use the Gauss–Laguerre quadrature. I am considering about substitution the lower limits. It looks like $$\int_{0}^{\infty}{{e^{-(x-10)}}f\left(x-10\...
3
votes
1
answer
88
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How should I prove $\int_0^\infty\frac{d^j}{dx^j}(x^je^{-x})dx=0$? [duplicate]
Context: Using the weighted inner product definition $$\langle f,g\rangle_{w(x)}=\int_a^bf(x)g(x)w(x)dx$$ for real valued functions $f(x),g(x),w(x)$, I wish to show that the following two functions ...
- 1,105
1
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0
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The Correct Change of Variable of an nth Order Derivative
The generalized Laguerre polynomials has the form:
$L_n^m(x):=\dfrac{x^{-m}e^x}{n!}\dfrac{\mathrm{d}^n}{\mathrm{d}x^n}(e^{-x}x^{n+m})$
My question is, what will be the $n^{th}$ order derivative when $...
0
votes
3
answers
95
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Integration of Gaussian $\times$ Laguerre functions
What is
$\begin{align}
\int_{-\infty}^{\infty}{e^{-ax^2+bx+c}L_n(dx^2+ex+f)dx}
\end{align}$ ?
Are there any identities that are close to this form that could be helpful?
2
votes
2
answers
142
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Generating function of the Confluent Hypergeometric Function of the First Kind
Let $x>0$ and $t\in(-1,1)$. Consider
$$\sum_{m=1}^\infty t^m \sum_{l=1}^m\binom{m-1}{l-1}\frac{(-x)^l}{l!}\,.$$
Can you find a closed expression for this series? Thank you for your time!
It reminds ...
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3
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0
answers
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Double series with Beta reciprocals $\sum_{j=0}^\infty \sum_{k=0}^\infty \frac{x^j}{j!}\frac{y^k}{k!} \frac{1}{\boldsymbol{B}(j+1,k+1)} = ? $
In my research I encountered the following double series involving reciprocals of Beta functions:
\begin{equation}
f(x,y) :=\sum_{j=0}^\infty \sum_{k=0}^\infty \frac{x^j}{j!}\frac{y^k}{k!} \frac{1}{\...
- 334
0
votes
0
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Laguerre polynomials and ill-defined Jacobi operators
I've been getting familiar with the theory of orthogonal polynomials, and one of the fundamental theorems that I'm working with states that a sequence of orthonormal polynomials $p_{n}(x)$ satisfies a ...
- 205
3
votes
1
answer
90
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Generating Functions and Associated Laguerre Polynomials
To give you context, I am currently attempting to derive the radial wavefunctions for a hydrogenic atom, from scratch.
B.H. Bransden, C.J. Joachain - Physics of Atoms and Molecules states:
$$U_{p}(\...
- 33
0
votes
0
answers
48
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Laguerre polynomial problem
It is known that
$\sum^{\infty}_{n=0}z^nL_n(x)=\frac{1}{1-z}e^{-xz/(1-z)}$
where $L_n(x)$ is the Laguerre polynomial.
It there any neat way of expressing the following term:
$\sum^{\infty}_{n=0}z^{n+m}...
- 11
3
votes
0
answers
72
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Combinatorial problem: triple binomial product related to squared Laguerre polynomials
Context
Hydrogenic wavefunctions [1] include a factor given by Laguerre polynomials [2]. These wavefunctions are often encountered in a first course in quantum mechanics. They also appear in ...
- 994
1
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0
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98
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Simple Identity for Derivative of Laguerre Polynomial
I'm working with Laguerre polynomials for numerically solving a differential equation, and I've stumbled upon an identity that I feel should be documented somewhere (e.g., https://en.wikipedia.org/...
- 787
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0
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How is the relation between the general Laguerre Differential equation and associated Laguerre differential equation deduced?
From Wikipedia, the Laguerre Differential Equation is defined as follows:
\begin{align}
x y'' + (v + 1 - x) y' + \lambda y = 0
\end{align}
By definition, the solution of this differential equation is ...
- 65
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votes
1
answer
157
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proving orthogonality of associated Laguerre polynomial using Generating function
I have been trying to prove the following orthogonal relation which is used for the normalization of the hydrogenic radial wave function,
$$\int_{0}^{\infty}e^{-\rho}\rho^{2l+2}[L^{2l+1}_{n+l}]^2d\rho=...
- 573
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0
answers
146
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Prove that the sequence of Laguerre polynomials is total
I'm trying to read chapter 3.7 from Kreyszig where he talks about Legendre, Hermite, and Laguerre polynomial. Here's how the definitions are -
Consider the space $L^2[0, \infty)$ with inner product $\...
0
votes
1
answer
571
views
Construct First 3 Orthogonal Polynomials with Gram-Schmidt
Consider the set of functions $u(x)=x^n,\,\,$ with $n=0,1,2, \dots$.
Use the Gram-Schmidt procedure to construct the first 3 orthogonal polynomials of:
$$\text{Laguerre:} \;\;\;\;L_n(x),\;\;\;\; \text{...
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1
vote
1
answer
228
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Computing the normalisation constant of the Laguerre polynomials
How does one compute the normalisation constant for the Laguerre polynomials from the Rodrigues formula, i.e. $\tfrac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}$? I tried:
$$
\int_0^\infty w(x)L_n^{(\alpha)}...
user185346
0
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1
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217
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Quadrature for logarithmic weight: $ \int_0^1 f(x) x \log x\, dx.$
Is there a standard way to evaluate (numerically) the integral
$$ \int_0^1 f(x) x \log(x) dx .$$
I was trying the substitution $u = -2\log(x)$, and then use Gauss-Laguerre quadrature. But it ...
0
votes
0
answers
335
views
Can an associated Laguerre polynomial be expressed in terms of a Bessel function of the first kind $J_a$?
I have found that associated Laguerre polynomials can be expressed in terms of spherical Bessel function ($j_n$, $y_n$) but what about in terms of Bessel functions of the first kind ($J_a$)?
The ...
2
votes
1
answer
98
views
Proof of an equation with the aid of Laguerre functions?
The generating function for the Laguerre functions {$\phi_m(x_3;\alpha)$} is:
\begin{align}\frac{\alpha ^{1/2}e^{-(1/2)\alpha x_3 (1+s)/(1-s)}}{1-s}= \sum_{m=0}^{\infty} s^m\phi_m(x_3;\alpha)\end{...
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1
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How to expand the product of Laguerre polynomials into a sum of series？
In the course of my research, I needed a formula and found it, but I can not understand the derivation process of the formula. How to extract the $t^n$ and get the $\theta(m-p)$ in the last step? Can ...
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0
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Can someone help with step by step method of finding the normalization constant of this wave function?
$$
\text R(x)= \text A x^{\left(\frac{\lambda+1}{2}\right)} e^{-\eta x / 2} \text F_{1}\left(-n, \lambda+\frac{3}{2}, x\right)
$$
where $\text A$ is normalization constant. Using the normalization ...
0
votes
1
answer
442
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Rodrigues' Formula for Laguerre equation
This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :
Starting from the Laguerre ODE,
$xy''+(1-x)y'+\lambda y =0 $,
obtain the Rodrigues formula for its polynomial ...
- 618
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votes
0
answers
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Limiting distribution of generalized derangement
Suppose there are $N$ people in a party. Each of them brings $k$ gifts. When the party is over, each of them takes $k$ gift randomly. Denote $T$ is the number of gifts return to its original giver. ...
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0
answers
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Proving Stability of a Function of Laguerre Polynomials
I'm trying to prove that the following potential is stable at its critical point:
$$
F_{\textbf{n}}(x) = x - \sum_{\ell=1}^{r} \ln G_{n_{\ell}}(x),
$$
where $\textbf{n} = (n_1, n_2, \ldots, n_r)$ and
$...
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votes
0
answers
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An Integral Equation for the Square of a Laguerre Polynomial
The following integral equation was presented back in the late 30's by Watson and Szego (Journal of the London Mathematical Society) but I cannot access the journal. Any ideas on a proof ?
$$e^{-x} x^{...
- 326
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votes
2
answers
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Proving that the Laguerre polynomials do indeed solve the differential equation
I am trying to show that the Laguerre differential equation, given in my homework problem as
$xL''_n(x)+(1−x)L'_n(x)+ nL_n(x) = 0$,
is indeed solved by the Laguerre polynomials in their closed sum ...
- 41
3
votes
1
answer
110
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Sum over (squares of) Laguerre Polynomials
I'm looking for a closed form of the sum \begin{equation}
\sum_{n=0}^\infty \frac{n!}{(n+k)!} (L_n^k(x))^2 t^n,
\end{equation}
where $L_n^k(x)$ are the Laguerre Polynomials. I have been looking for ...
- 31
1
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0
answers
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A modification of the Laguerre product expansion
Given a product of Laguerre polynomials, $L_n(x) L_m(x)$, a particular question to ask is the expansion of this product in terms of the Laguerre polynomials $\{L_i(x)\}$ themselves. That is, we would ...
- 123
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votes
3
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Is the sum of the first N Laguerre polynomials (with alternating signs) always positive?
I have noticed that the following simple sum of Laguerre polynomials (weighted with alternating signs) seems to be positive for any $N$ when $x>0$:
$$\sum_{k=0}^{N}\;(-1)^{k}\;L_{k}(x)$$
More ...
- 133
1
vote
1
answer
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Prove $\int_0^\infty e^{-x} x^k [L^k_n (x)]^2 \, dx=\frac{(n+k)!}{n!}$
How can I prove the normalization ratio of associated Laguerre polynomials: $$\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
1
answer
456
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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
1
answer
172
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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 ...
- 109
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vote
0
answers
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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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0
answers
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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 ...
- 303
2
votes
2
answers
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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 ...
- 7,495
1
vote
1
answer
111
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 ...
- 287
1
vote
1
answer
200
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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 $f$ ...
- 325
2
votes
2
answers
202
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 ...
- 363
2
votes
0
answers
205
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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 ...
- 21
2
votes
1
answer
400
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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:
$$
\sum\limits_{k=0}^{n}C_n^k(-...
- 21
3
votes
1
answer
328
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Integral relation between Hermite and Laguerre polynomials
I'd like to proove the following integral relation
$$ \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\...
- 33