legendres polynomial and recurrence formula or rodrigues method

Question

im not being able to do the below sum....tried with recurrence and all other methods not coming please help The question is given below:- P'n+1 +P'n= P0+3P1 +5P2 +.....+(2n+1)Pn where Pn= legendres polynomial

Answer 1

Use the Legendre polynomial's property $$P_{m+1}' - P_{m-1}' = (2m+1)P_m$$ Summing, we get $$\sum_{m=1}^{n} (2m+1)P_m = \sum_{m=1}^{n} (P_{m+1}' - P_{m-1}') = P_{n+1}'+P_{n}'-P_{1}'-P_{0}'= P_{n+1}'+P_{n}'-1$$ As $P_0=1$, this can be rewritten as $$P_{n+1}'+P_{n}' = \sum_{m=0}^{n} (2m+1)P_m$$