Questions tagged [laguerre-polynomials]

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

Filter by
Sorted by
Tagged with
2 votes
0 answers
44 views

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
0 votes
1 answer
67 views

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
1 vote
1 answer
41 views

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
1 vote
1 answer
292 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)...
user2224350's user avatar
1 vote
0 answers
46 views

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
0 votes
1 answer
37 views

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
  • 469
1 vote
0 answers
35 views

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
  • 256
3 votes
0 answers
68 views

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
  • 3,164
1 vote
1 answer
79 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
  • 55
0 votes
0 answers
113 views

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
88 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 ...
Alborz's user avatar
  • 1,105
1 vote
0 answers
29 views

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
0 votes
3 answers
95 views

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
142 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 ...
ffff's user avatar
  • 41
3 votes
0 answers
133 views

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
  • 334
0 votes
0 answers
20 views

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 ...
miggle's user avatar
  • 205
3 votes
1 answer
90 views

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
0 votes
0 answers
48 views

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
72 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
1 vote
0 answers
98 views

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
  • 787
1 vote
0 answers
54 views

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
0 votes
1 answer
157 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
  • 573
1 vote
0 answers
146 views

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
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{...
James's user avatar
  • 11
1 vote
1 answer
228 views

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)}...
user avatar
0 votes
1 answer
217 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
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 ...
Curious One's user avatar
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{...
likelee's user avatar
  • 145
2 votes
1 answer
135 views

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
  • 145
1 vote
0 answers
46 views

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
0 votes
1 answer
442 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
  • 618
0 votes
0 answers
25 views

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
25 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
197 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
793 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
110 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
60 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
  • 123
10 votes
3 answers
248 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
  • 133
1 vote
1 answer
312 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, $$\...
Rebeca Lie Yatsuzuka Silva's user avatar
1 vote
1 answer
456 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
4 votes
1 answer
172 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 ...
Almhz's user avatar
  • 109
1 vote
0 answers
45 views

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

$$L_n^k(x)= \cfrac{e^xx^{-k}}{n!}\cfrac{d^n}{dx^n}(e^{-x}x^{n+k})$$
OscarR's user avatar
  • 41
2 votes
0 answers
67 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 ...
Samantha's user avatar
  • 303
2 votes
2 answers
2k 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 ...
Sooraj S's user avatar
  • 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 ...
Teun's user avatar
  • 287
1 vote
1 answer
200 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 $f$ ...
Plussoyeur's user avatar
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 ...
Samuel Johnston's user avatar
2 votes
0 answers
205 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 ...
Flip9's user avatar
  • 21
2 votes
1 answer
400 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(-...
BUAA_Wander's user avatar
3 votes
1 answer
328 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\...
Marie's user avatar
  • 33