In Maple, one can type simplify(LegendreP(2*n-2,0)-LegendreP(2* \(\mathbf{n}, \mathbf{0})\) ); to find a value for \(P_{2 n-2}(0)-P_{2 n}(0)\). It gives the result in terms of Gamma functions. However, in Example 6. 10 for Fourier-Legendre series, the value is given in terms of double factorials! So, we have

\[P_{2 n-2}(0)-P_{2 n}(0)=\frac{\sqrt{\pi}(4 n-1)}{2 \Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}=(-1)^{n-1} \frac{(2 n-3)!!}{(2 n-2)!!} \frac{4 n-1}{2 n}\]

You will verify that both results are the same by doing the following:

a. Prove that \(P_{2 n}(0)=(-1)^{n} \frac{(2 n-1)!!}{(2 n)!!}\) using the generating function and a binomial expansion.

b. Prove that \(\Gamma\left(n+\frac{1}{2}\right)=\frac{(2 n-1)!!}{2^{n}} \sqrt{\pi}\) using \(\Gamma(x)=(x-1) \Gamma(x-\) \(1)\) and iteration.

c. Verify the result from Maple that \(P_{2 n-2}(0)-P_{2 n}(0)=\frac{\sqrt{\pi}(4 n-1)}{2 \Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}\).

d. Can either expression for \(P_{2 n-2}(0)-P_{2 n}(0)\) be simplified further?