Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 1 / b

Solution

(x w)^{'} = (1 - x) w

. Thus

x w^{'} = - x w

, and the constant factor can be chosen so that

w (x) = e^{- x}, - (x e^{- x} y^{'})^{'} = n e^{- x} y .

This fixes the otherwise arbitrary normalization of the weight consistently with the requested unit norms.

p_{n} (x) = \sum_{j = 0}^{n} c_{j} x^{j}

. Comparing coefficients gives

(j + 1)^{2} c_{j + 1} + (n - j) c_{j} = 0, c_{0} = 1.

Consequently

c_{j} = (- 1)^{j} (j n) / j!

, identifying the solutions as Laguerre polynomials. In particular,

p_{0} = 1, p_{1} = 1 - x, p_{2} = 1 - 2 x + \frac{x ^{2}}{2} .

\int_{0}^{\infty} e^{- x} x^{j} d x = j!

. Therefore

∥ p_{0} ∥_{w}^{2} ∥ p_{1} ∥_{w}^{2} ∥ p_{2} ∥_{w}^{2} = 1, = 1 - 2 (1!) + 2! = 1, = 1 - 4 (1!) + 5 (2!) - 2 (3!) + \frac{1}{4} (4!) = 1.

For later use, integration by parts in the weighted eigenvalue equations also gives

(n - m) ⟨ p_{n}, p_{m} ⟩_{w} = [x e^{- x} (p_{n} p_{m}^{'} - p_{n}^{'} p_{m})]_{0}^{\infty} = 0.

The endpoint expression vanishes because of the factor

x

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