Using the formula below to obtain Legendre polynomial $P_2(x),P_3(x) $.
$$P_n(x) = \begin{cases} 1, & \text{ if } n = 0; \\ & \\ x, & \text{ if } n = 1; \\ & \\ \dfrac{1}{n}[(2n-1)xP_{n-1}(x)+(n-1)P_{n-2}(x)], & \text{ if } n \geq 2. \end{cases}$$
My attempt:
$$P_2(x) = \dfrac{1}{2}\left(3xP_1(x) + P_0(x)\right) = \dfrac{1}{2}\left(3x\times x + 1\right) = \dfrac{1}{2}\left(3x^2 + 1 \right)$$
$$P_3(x) = \dfrac{1}{3}\left(5xP_2(x) + 2P_1(x)\right)= \dfrac{1}{2}\left(5x^3 + 3x \right)$$
But the actual result should be $P_2(x) = \dfrac{1}{2}(3x^2-1)$ and $P_3(x) = \dfrac{1}{2}(5x^3 - 3x )$.
What have I done incorrectly to the sign?