Chapter 12.23, Problem 27MP

Show that $R=lx-\left(1-x^2\right)D$ and $L=lx+\left(1-x^2\right)D,$ where $D=d/dx,$ are raising and lowering operators for Legendre polynomials [compare Hermite functions, (22.1) to (22.11) and Bessel functions, Problems 22.29 and 22.30]. More precisely, showthat $R{P}_{l-1}(x)=l{P}_l(x)$ and $L{P}_l(x)=l{P}_{l-1}(x).$ Hint: Use equations (5.8d) and (5.8f). Note that, unlike the raising and lowering operators for Hermite functions, here $R$ and $L$ depend on $l$ as well as x, so you must be careful about indices. The $L$ operator operates on $P_1$, but the $R$ operator as given operates on $P_{l-1}$ to produce $l{P}_1$.

[If you prefer, you could replace $l$ by $l+1$ to rewrite $R$ as $(l+1)x-\left(1-x^2\right)D$; then it operates on Assuming that all $P_l(1)=1,$ solve $L{P}_0(x)=0$ to find $P_0(x)=1,$ and then use raising operators to find $P_1(x)$ $P_2(x)$.