# Questions tagged [laguerre-polynomials]

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

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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 ...
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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: $$\text{Laguerre:} \;\;\;\;L_n(x),\;\;\;\; \text{... 1 vote 1 answer 228 views ### Computing the normalisation constant of the Laguerre polynomials How does one compute the normalisation constant for the Laguerre polynomials from the Rodrigues formula, i.e. \tfrac{\Gamma(n+\alpha+1)}{n!}\delta_{n,m}? I tried:$$ \int_0^\infty w(x)L_n^{(\alpha)}... 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 ...
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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 ...
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. ...