Chapter 7.4, Problem 65E

(a)    Laguerre’s differential equation

$t{y}^{″}+\left(1-t\right)y^{\prime }+ny=0$

is known to possess polynomial solutions when n is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by L_n(t). Find $y=L_n\left(t\right)$, for n = 0, 1, 2, 3, 4 if it is known that $L_n\left(0\right)=1$.

(b)    Show that

$L\left\{\frac{e^t}{n!}\frac{d^n}{d{t}^n}{t}^n{e}^{-t}\right\}=Y\left(s\right),$

where $Y\left(s\right)=L\left(y\right)$ and $y=L_{}\left(t\right)$ is a polynomial solution of the DE in part (a). Conclude that

$L_n\left(t\right)=\frac{e^t}{n!}\frac{d^n}{d{t}^n}{t}^n{e}^{-t},n=0,1,2,\dots$

This last relation for generating the Laguerre polynomials is the analogue of Rodrigues’ formula for the Legendre polynomials. See (36) in Section 6.4.

Rodrigues’ formula for these polynomials is

$P_n\left(x\right)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}{\left(x^2-1\right)}^n,n=0,1,2,\dots$    (36)