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Questions tagged [laguerre-polynomials]

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

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Convergence rate of Laguerre coefficients for polynomially bounded functions

Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies: $$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$ for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
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Integral of a Generalized Laguerre Polynomial [closed]

I am looking for the solutions to the following integral: $$ I_{n} = \int_{0}^{\infty}x^{4} \operatorname{L}_{n}^{3}\left(x\right) {\rm e}^{-\left(n + 3\right)x/2}\,{\rm d}x,\qquad n \in\mathbb{N}_{0} ...
Rocky's user avatar
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Calculate Components of square integrable functions w.r.t. some basis

Consider the space of square integrable functions on the non negative real numbers $L^2(\mathbb{R}_0^+)$. I found out, that the Laguerre functions modulo some normalization define an orthonormal basis ...
Aralian's user avatar
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A quadrature rule given for this $\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) + a_3 f(t_3) + a_4 f(t_4) + R(f)$ using Gauss-Laguerre? [closed]

Can a Laguerre polynomial be used for this problem? How does $f'(0)$ square in? Find coefficient and nodes for the following quadrature formula. $\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) +...
Doru Popa's user avatar
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Symmetrized versions of Laguerre polynomials

It is well known that the Laguerre polynomials are complete in the Hilbert space $L^2$ with inner product $$ \langle f,g\rangle = \int_0^{\infty} e^{-x} f(x) g(x) dx $$ In general, these polynomials ...
Jonathan's user avatar
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Relation between Laguerre function and confluent hypergeometric function

The Wikipedia article on the Rice probability distribution says that \begin{equation} L_{q}(x)=L_{q}^{(0)}(x)=M(-q,1,x)=\,_{1}F_{1}(-q;1;x). \end{equation} where $L_q(x)$ is a Laguerre function (with ...
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Quantization Laguerre equation

I saw this interesting article https://www.mdpi.com/2073-8994/14/4/741 about how the quantization of the eigenvalues of the Legendre equation is a consequence of parity symmetry and imposing that the ...
Marc Navarro's user avatar
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Limiting behavior of the associated Laguerre polynomial

From How to show orthogonality of associated Laguerre polynomials? or Finding a generating function for the Laguerre polynomials it appears that the expression, $L^q_p(x)$ is that of an associated ...
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Simple Laguerre polynomial for half-integer orders

I am trying to calculate the simple Laguerre polynomial for the half-integer orders. From https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch7/laguerre.html, I can get the Laguerre ...
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Approximate solution of the following integral

I have the following integral. $$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$ with $$ \alpha \in \mathbb{R}^+, \quad q, b, c \in \mathbb{R}, \quad j = \sqrt{-1}$$ I have ...
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Finding demonstration of a formula used for calculating Laguerre integrals

does someone know where i can find the mathematical demonstration of this formula? http://functions.wolfram.com/05.08.21.0009.01 I honestly dont even know what to search for on the internet, seems a ...
The Operator's user avatar
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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{...
First Last's user avatar
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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 $L_{n-l-1}^{2l+1}(2p)$...
AtomProgrammer's user avatar
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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 ...
Andras Vanyolos's user avatar
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1 answer
323 views

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)...
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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}^{\...
Tobias Wolf's user avatar
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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^...
Galilean's user avatar
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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 $\...
Dasi's user avatar
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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 ...
Math Attack's user avatar
1 vote
1 answer
144 views

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. ...
Ivan R.'s user avatar
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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\...
Michael M's user avatar
3 votes
1 answer
112 views

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 ...
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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 $...
Ozan Turhan Gündüz's user avatar
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3 answers
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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?
Saurabh Shringarpure's user avatar
2 votes
2 answers
178 views

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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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}{\...
ARedder's user avatar
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1 answer
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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}(\...
lethobentho's user avatar
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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}...
Liu Long's user avatar
3 votes
0 answers
78 views

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 ...
Michael Levy's user avatar
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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/...
superckl's user avatar
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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 ...
Prince Khan's user avatar
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1 answer
172 views

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=...
seraphimk's user avatar
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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 $\...
Aniket Bhattacharyea's user avatar
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1 answer
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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{...
James's user avatar
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1 vote
1 answer
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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)}...
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301 views

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 ...
Vicente GOMEZ HERRERA's user avatar
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492 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 ...
Curious One's user avatar
2 votes
1 answer
100 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{...
likelee's user avatar
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2 votes
1 answer
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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 ...
likelee's user avatar
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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 ...
tolulope ojuola's user avatar
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1 answer
504 views

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 ...
Physmath's user avatar
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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. ...
Oolong Milktea's user avatar
1 vote
0 answers
26 views

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 $...
motherboard's user avatar
4 votes
0 answers
232 views

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^{...
Joel Storch's user avatar
4 votes
2 answers
933 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 ...
Trang Nguyen's user avatar
3 votes
1 answer
132 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 ...
Stephphen's user avatar
1 vote
0 answers
70 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 ...
K L's user avatar
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10 votes
3 answers
280 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 ...
Lucky's user avatar
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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, $$\...
Rebeca Lie Yatsuzuka Silva's user avatar
1 vote
1 answer
522 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 ...
Tuong Nguyen Minh's user avatar