Questions tagged [laguerre-polynomials]

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

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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 ...
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Calculating the Laguerre function

In the Noncentral Chi Distribution Wikipedia page, the calculated Mean is: $$ {\sqrt{{\pi\over2}}L_{1/2}^{(k/2-1)} \left( {\small{-\lambda^2\over2}}\right)}$$ I am calculating the average distance ...
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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=...
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Connection between generating functions with special polynomials

I was learning special function (ODE II course) where I encounter various kind of special polynomials like Legendre, Bessel's, Hermite and Laguerre. And many of their properties (specially recursive ...
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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 $\...
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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{...
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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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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 ...
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Hyperbolic equation and non-integer order Laguerre polynomials

I am currently working on a cosmology problem and found the following equation $$ zy^{\prime\prime}(z) + (1+\Delta-z) y^\prime(z) - \alpha y(z) = 0 $$ where $\Delta$ and $\alpha$ are rational numbers ...
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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 ...
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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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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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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 ...
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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 ...
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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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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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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^{...
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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 ...
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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 ...
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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 ...
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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 ...
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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, $$\...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 $f$ ...
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2 votes
2 answers
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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 ...
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2 votes
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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 ...
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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(-...
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3 votes
1 answer
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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\...
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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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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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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 $\sigma (x)=e^{-x}$. ...
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11 votes
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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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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 ...
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2 votes
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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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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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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....
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