Chapter 11 - Series Solutions to Linear Differential Equations - 11.3 The Legendre Equation - Problems - Page 749: 3

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Use Rodrigues’ formula to determine the Legendre polynomial of degree 3: $P_3(x)=\frac{1}{3!2^3}\frac{d^3}{dx^3}(x^2-1)^3\\ =\frac{1}{3!2^3}\frac{d^2}{dx^2}[6x(x^2-1)^2]\\ =\frac{1}{3!2^3}\frac{d}{dx}[6(5x^4-6x^2+1)]\\ =\frac{1}{3!2^3}[24x(5x^2-3)]\\ =\frac{1}{2}x(5x^2-3)$