Chapter 16, Problem 20PS

Laguerre’s Equation Consider Laguerre’s Equation

$xy\text{'}\text{'}+\left(1-x\right)y\text{'}+ky=0$.

Polynomial solutions of Laguerre’s equation are called Laguerre polynomials and are denoted by $L_k\left(x\right)$. Use a power series of the form

$y={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{a}_n{x}^n}$

to show that

$L_k\left(x\right)={\displaystyle \underset{n=0}{\overset{k}{\sum }}\frac{{\left(-1\right)}^{n}k!x^n}{\left(k-n\right)!{\left(n!\right)}^2}}$

Assume that $a_0=1$.

Determine the Laguerre polynomials $L_0\left(x\right),L_1\left(x\right),L_2\left(x\right),L_3\left(x\right)\text{, and }{L}_4\left(x\right)$.