Laguerre Polynomials
Definition
$\displaystyle L_{n} := {{ e^{x} } \over { n! }} {{ d^{n} } \over { dx^{n} }} \left( e^{-x} x^{n} \right)$ is called Laguerre Polynomial.
Basic Properties
Recursion Formula
- [0]: $$L_{n+1} (x) = {{ 1 } \over { n+1 }} \left[ \left( 2n + 1 - x \right) L_{n} (x) - n L_{n-1} (x) \right]$$
Orthogonal Set
- [1] Inner Product of Functions: Given the weight $w$ as $\displaystyle w(x) := e^{-x}$ for $\displaystyle \left<f, g\right>:=\int_a^b f(x) g(x) w(x) dx$, $\left\{ L_{0} , L_{1}, L_{2}, \cdots \right\}$ becomes an Orthogonal Set.
Description
The Laguerre Polynomial for $n = 0, \cdots , 3$ is represented as follows. $$ \begin{align*} L_{0} (x) =& 1 \\ L_{1} (x) =& -x + 1 \\ L_{2} (x) =& {{1} \over {2}} \left( x^{2} - 4x + 2 \right) \\ L_{3} (x) =& {{1} \over {6}} \left( - x^{3} + 9 x^2 -18x + 6 \right) \end{align*} $$ The Laguerre Polynomial is also defined as a solution of the Laguerre Differential Equation $xy’’ + (1-x) y ' + ny = 0$.
Unfortunately, the Closed Form for the Laguerre Nodes $x_{k}$ that satisfy $L_{n} ( x_{k} ) = 0$ is not known, and it is currently calculated numerically. Notably, in [1], $\left\{ L_{0} , L_{1}, L_{2}, \cdots \right\}$ is not only an Orthogonal Set but also an orthonormal set.