If $P_n(x)$ is a Legendre polynomial of degree $n$. If a is such that $P_n(a)=0$ that is $a$ is a root of $P_n(x)=0$. Then the $P_{n-1}(a)$ and $P_{n+1}(a)$ is: equal ? or not equal ? Or are of opposite signs ? Or are of the same signs?
Which one is correct?
I think " not equal " is correct
Hint: The Legendre polynomials satisfy the following recurrence relation (by Bonnet) $$ (n+1)P_{n+1}(x)=(2n+1)xP_n(x)-nP_{n-1}(x). $$