Chapter 12.22, Problem 15P

Solve the Laguerre differential equation $x{y}^{\prime \prime }+(1-x)y^{\prime }+py=0$ by power series. Show that the $a_0$ series terminates if $p$ is an integer. Thus for each integer n, the differential equation (22.21) has one solution which is polynomial of degree n. These polynomials with $a_0=1$ are the Laguerre polynomials $L_n(x).$ Find $L_0(x),L_1(x),L_2(x),$ and $L_3(x).$ (This is an eigenvalue problem - compare Problem 5 and Section 2.)