Chapter 12.10, Problem 3P

Show that the functions $P_l^m(x)$ for each $m$ are a set of orthogonal functions on $(-1,1),$ that is, show that ${\displaystyle {\int }_{-1}^1{P}_l^m}(x)P_n^m(x)dx=0,l\ne n$.

Hint: Use the differential equations (10.1) and follow the method of Section 7.