Word Count : 260

---

We are asked to show that [ \int_{-1}^{1} x^2 P_{n-1}(x) P_{n+1}(x) , dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)} ] where (P_n(x)) are Legendre polynomials. We solve this step by step using known recurrence relations and orthogonality properties. 1. Use the recurrence relation for Legendre polynomials: [ x P_n(x) = \frac{n+1}{2n+1} P_{n+1}(x) + \frac{n}{2n+1} P_{n-1}(x) ] 2. Express (x^2 P_{n-1}(x)) in terms of Legendre polynomials: First, write (x P_{n-1}(x)) _________ _______ ________ _____ _________.
_____ ________ ___ ____ ____ _________ ________ ________ ____.
___ ___ _______ ___ ____ ____ ________ ______ ____.
________ __________ _________ ________ _____ ________ ______.
____ _______ ____ __________ __________ _________ _____ ____ ___.
__________ __________ ________ _________ ____ ___ ____ __________ _________ ____ _______.
___ ______ ___ ________ ________ ________ ________ ______.
_______ ____ ___ __________ _____.
___ __________ ________ _________ ______ _________.
____ ____ __________ ________ ______ _______.
______ _____ _____ ________ ___ ____ ___.
________ _______ _________ ________ _______.
_____ ________ __________ ________ ________ ___ __________.
_______ _________ ___ _____ __________ _______ _______.
___ ___ _______ _________ _________ _____ ___ ____ _____ _____ ____ ________.
________ _____ ________ ____ _____.
______ ____ _________ ____ ________ __________ __________.
_______ _________ _____ _________ _________.
___ _______ __________ _______ _____.
_____ __________ __________ ___ ________ ____ _________ ________ ______ _______.
________ _____ __________ __________ _______ _________ ___ _________.
___ ___ ______ __________ __________ ______ __________ ______ ___ ________.
________ ___ ______ _______ ______.
_______ __________ __________ _____ _____ ____.
______ _____ _______ ____ _________ ____.
_______ __________ __________ _________ _____ ____ _________ ______ _________.
___ ________ _______ ____ __________ ___.
Get Full Answer on WhatsApp