Chapter 12.22, Problem 29P

Show that $R_p=\frac{p}{x}-D$ and $L_p=\frac{p}{x}+D$ where $D=d/dx,$ are raising and lowering operators for Bessel functions, that is, show that $R_p{J}_p(x)=J_{p+1}(x)$ and $L_p{J}_p(x)=$ $J_{p-1}(x).$ Hint: Use equations (15.5). Note that these operators depend on $p$ as well asx, so they are not as simple as the Hermite function raising and loweringoperators (22.7) and ( 22.8 ). If you want to operate, say, on $J_{p+1}$, you must change $p$ in $R$ or $L$ to $p+1$, etc. Making this adjustment, show that the equations $LR{J}_p=J_p$ and $RL{J}_p=J_p$ both give Bessel’s equation.