The Jacobi orthogonal polynomials (P_{n}^{alpha, beta}(x)), which are the solutions of the ode [ left(1-x^{2} ight) frac{d^{2}

The Jacobi orthogonal polynomials \(P_{n}^{\alpha, \beta}(x)\), which are the solutions of the ode

\[ \left(1-x^{2}\right) \frac{d^{2} P_{n}^{\alpha, \beta}}{d x^{2}}+[\beta-\alpha-(\alpha+\beta+2) x] \frac{d P_{n}^{\alpha, \beta}}{d x}+n(n+\alpha+\beta+1) P_{n}^{\alpha, \beta}=0 \]

are related to the Sturm-Liouville ode, which in turn appears in the construction of basis functions and the evaluation of certain functional integrals. Hence, clarify this relation by determining the eigenvalues and eigenfunctions of the Sturm-Liouville operator (6.35)

\[ \mathcal{L} \equiv \frac{1}{w(x)}\left[\frac{d}{d x}\left(p(x) \frac{d}{d x}\right)-q(x)\right] \]

defined on the interval \([-1,1]\) for \(p(x)=(1-x)^{\alpha+1}(1+x)^{\beta+1}, q(x)=0\) and weight function \(w(x)=(1-x)^{\alpha}(1+x)^{\beta}\). Assume \(\alpha, \beta>-1\).

(4.1.1) Show that the Sturm-Liouville operator is self-adjoint, i.e. \((\mathcal{L} u, v)=\) \((u, \mathcal{L} v)\) holds with respect to the scalar product

\[ (u, v) \equiv \int_{-1}^{1} d x w(x) u(x) v(x) \]

for \(u(x), v(x) \in C_{[-1,1]}^{2}\).

(4.1.2) Verify that the Jacobi polynomials are eigenfunctions of the SturmLiouville operator for the specified coefficients.

(4.1.3) Expand the test function \(f(x)=\sin (\pi k x)\) for \(k=1\) and \(k=16\) in terms of the Jacobi basis \(\left\{P_{n}^{\alpha, \beta}, n=0,1,2, \ldots\right\}\) for \(\alpha=\beta=0\) fixed. Compute and plot the error

\[ e(N) \equiv\left|f(x)-\sum_{i=0}^{N} c_{i} P_{i}^{\alpha, \beta}\right| \]

as function of \(N\) and the test function \(f(x)\) together with partial sums of the expansion.