Chapter 12.23, Problem 5MP

Show that ${\displaystyle \underset{0}{\overset{n}{\sum }}(2l+1)}{P}_l(x)=P_n^{\prime }(x)+P_{n+1}^{\prime }(x).$ Hint: Use mathematical induction as follows:

(a) Verify the formula for $n=0$

(b) Assuming that the formula is true for $l=n-1,$ show [using (5.8e)] that it is true for $l=n$.