Laguerre Polynomials
Definition
The Laguerre polynomial is defined by the following methods.
As a Solution to a Differential Equation
A solution to the Laguerre differential equation given below is called a Laguerre polynomial.
$$ xy^{\prime \prime} + (1-x)y^{\prime} + ny = 0, \quad n=0,1,2,\cdots $$
Rodrigues’ Formula
The function $L_{n}$ given below is called the Laguerre polynomial.
$$ L_{n}(x) = \frac{1}{n!}e^{x}\frac{ d ^{n}}{ dx^{n} }(x^{n}e^{-x}) \tag{1} $$
The above formula is called Rodrigues’ formula.
Explanation
By definition, $L_{n}$ is indeed a polynomial ‘function’, but by convention it is called a Laguerre ‘polynomial’. This is not only the case in Korean; the English expression is also Laguerre polynomial rather than polynomial function.
By $(1)$, we can see that $L_{n}$ is a polynomial of degree $n$. The first few Laguerre polynomials are as follows.
$$ \begin{align*} L_{0}(x) &= 1 \\ L_{1}(x) &= -x+1 \\ L_{2}(x) &= \frac{1}{2}\left( x^{2}-4x+2 \right) \\ L_{3}(x) &= \frac{1}{6}\left( -x^{3}+9x^{2}-18x+6 \right) \\ \vdots & \end{align*} $$
The roots of the Laguerre polynomial are used as nodes for computing improper integrals in numerical analysis.