# Rodrigues' Formula for Laguerre equation

• I
• appmathstudent
In summary, Arfken's Mathematical Methods for Physicists 7th edition has a formula for polynomial solutions to the Laguerre ODE, which is stated as:\frac{1}{w(x)}(\frac{d}{dx})^n[w(x)p(x)^n]$$where ##w(x)## is an arbitrary function.w(x) = e^{-x} and ##L_n(x)## is given by e^x (\frac{d}{dx})^n[e^{-x}x^n]$$ where ##x^n## is a solution.The factor of ##\frac 1 {n!}##
appmathstudent
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

Delta2
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.

PeroK and appmathstudent
Thank you very much! I think I got it now

## 1. What is Rodrigues' Formula for the Laguerre equation?

Rodrigues' Formula is a mathematical formula used to solve the Laguerre equation, which is a special type of differential equation. It was developed by the French mathematician Olinde Rodrigues in the 19th century.

## 2. How is Rodrigues' Formula derived?

Rodrigues' Formula is derived using a combination of techniques from calculus, linear algebra, and special functions. It involves expanding a given function into a series of orthogonal polynomials and then using recursive relationships to simplify the equation.

## 3. What is the significance of Rodrigues' Formula?

Rodrigues' Formula is significant because it provides a general solution to the Laguerre equation, which is used in a variety of fields such as physics, engineering, and statistics. It allows for the efficient calculation of the solutions to this equation, which can be difficult to solve using other methods.

## 4. Can Rodrigues' Formula be applied to other types of equations?

While Rodrigues' Formula is specifically designed for the Laguerre equation, it can also be applied to other types of differential equations, such as the Hermite equation and the Legendre equation. However, the formula may need to be modified slightly to fit the specific equation.

## 5. Are there any limitations to using Rodrigues' Formula?

One limitation of Rodrigues' Formula is that it may not always provide a closed-form solution to the Laguerre equation. In some cases, the solution may need to be approximated using numerical methods. Additionally, the formula may not be applicable to equations with complex or non-integer coefficients.

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