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Laguerre Polynomials 📂Numerical Analysis

Laguerre Polynomials

Definition

Ln:=exn!dndxn(e−xxn)\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]: Ln+1(x)=1n+1[(2n+1−x)Ln(x)−nLn−1(x)]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 ww as w(x):=e−x\displaystyle w(x) := e^{-x} for <f,g>:=∫abf(x)g(x)w(x)dx\displaystyle \left<f, g\right>:=\int_a^b f(x) g(x) w(x) dx, {L0,L1,L2,⋯ }\left\{ L_{0} , L_{1}, L_{2}, \cdots \right\} becomes an Orthogonal Set.

Description

The Laguerre Polynomial for n=0,⋯ ,3n = 0, \cdots , 3 is represented as follows. L0(x)=1L1(x)=−x+1L2(x)=12(x2−4x+2)L3(x)=16(−x3+9x2−18x+6) \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=0xy’’ + (1-x) y ' + ny = 0.

Unfortunately, the Closed Form for the Laguerre Nodes xkx_{k} that satisfy Ln(xk)=0L_{n} ( x_{k} ) = 0 is not known, and it is currently calculated numerically. Notably, in [1], {L0,L1,L2,⋯ }\left\{ L_{0} , L_{1}, L_{2}, \cdots \right\} is not only an Orthogonal Set but also an orthonormal set.