 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/17b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Suppose

m < n

n

integrations by parts give

⟨ p_{n}, p_{m} ⟩ = (- 1)^{n} \int_{0}^{\infty} p_{m} (x) \frac{d ^{n}}{d x ^{n}} (x^{n + 1/2} e^{- x}) d x = \int_{0}^{\infty} p_{m}^{(n)} (x) x^{n + 1/2} e^{- x} d x = 0,

because

p_{m}^{(n)} = 0

. Every boundary term vanishes: the exponential controls infinity, while the remaining power of

x

has positive exponent at zero. Symmetry of the inner product gives orthogonality whenever

m \neq = n

For the norm, monicity gives

p_{n}^{(n)} = n!

, so the same calculation yields

⟨ p_{n}, p_{n} ⟩ = n! \int_{0}^{\infty} x^{n + 1/2} e^{- x} d x = n! Γ (n + \frac{3}{2}) .

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