I Rodrigues' Formula for Laguerre equation

appmathstudent
Messages
6
Reaction score
2
TL;DR Summary
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 solutions $$L_n (x)$$

According to Arfken (equation 12.9 ,chapter 12) the Rodrigues formula is :

$$ y_n(x) = \frac {1}{w(x)}(\frac{d}{dx})^n[w(x)p(x)^n]$$

I found that $$w(x) = e^{-x}$$ and then :

$$L_n(x) = e^x (\frac{d}{dx})^n[e^{-x}x^n]$$

But the answer is ,according to Arfken and everywhere else I look,is :

$$L_n(x)=\frac{e^x}{n!}.\frac{d^n}{dx^n}(x^ne^{-x})$$

I can't figure out exactly how $$ \frac{1}{n!}$$ appeared.
I think it might be related to the fact that $$ L_n(x) =\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{k!} \quad $$

Any help will be appreciated , thank you
 
Mathematics news on Phys.org
You can have any constant factor you want and the result is still a solution. The factor of ##\frac 1 {n!}## may simply be the convention.
 
The ##\frac{1}{n!}## is the normalisation constant, it ensures that:

\begin{align*}
\int_0^\infty e^{-x} L_n (x) L_n (x) dx = 1
\end{align*}

Explicitly, it ensures that:

\begin{align*}
\frac{1}{(n!)^2} \int_0^\infty e^{x} \frac{d^n}{dx^n} (x^n e^{-x}) \frac{d^n}{dx^n} (x^n e^{-x}) dx = 1
\end{align*}

This can be verified by using Leibnitz and some integration by parts:

\begin{align*}
& \frac{1}{(n!)^2} \int_0^\infty \left( \sum_{k=0}^n (-1)^{n-k} \frac{n!}{k! (n-k)!} x^{n-k} \right) \frac{d^n}{dx^n} (x^n e^{-x}) dx
\nonumber \\
& = \frac{1}{n!} \int_0^\infty x^n e^{-x} dx
\nonumber \\
& = \frac{1}{n!} n! = 1 .
\end{align*}

I'll leave you to fill in the details.
 
  • Like
Likes PeroK and appmathstudent
Thank you very much! I think I got it now
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top