1. (12 points) Let Pn (x) denote the nth order Legendre polynomial and recall 2 In 2n +1 = dis -1 EL P.(0)2 - .- with Ln

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1 12 Points Let Pn X Denote The Nth Order Legendre Polynomial And Recall 2 In 2n 1 Dis 1 El P 0 2 With Ln 1 (76.82 KiB) Viewed 32 times

1. (12 points) Let Pn (x) denote the nth order Legendre polynomial and recall 2 In 2n +1 = dis -1 EL P.(0)2 - .- with Ln sPn(s)Pn-1(s) ds. (a) Using our work from class, show that 2n (2n – 1)(2n + 1) Ln = + - (b) Prove the recurrence relation (n + 1)Pn+1(x) – (2n + 1)«Pn(x) + nPn-1(x) = 0. Hint: You may assume that there are real sequences An and Bn such that xPn (2) = An Pn+1(x) + Br Pn-1(2) for every x E R and n e N and use the result of part (a). = -1