Paper 1, Section I, B

Numerical Analysis
Part IB, 2011

Orthogonal monic polynomials p0,p1,,pn,p_{0}, p_{1}, \ldots, p_{n}, \ldots are defined with respect to the inner product p,q=11w(x)p(x)q(x)dx\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x, where pnp_{n} is of degree nn. Show that such polynomials obey a three-term recurrence relation

pn+1(x)=(xαn)pn(x)βnpn1(x)p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x)

for appropriate choices of αn\alpha_{n} and βn\beta_{n}.

Now suppose that w(x)w(x) is an even function of xx. Show that the pnp_{n} are even or odd functions of xx according to whether nn is even or odd.