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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 17B / c / Solution

For monic orthogonal polynomials, taking the inner product of the recurrence with

p_{n - 1}

gives

β_{n} = \frac{⟨ x p _{n} , p _{n - 1} ⟩}{⟨ p _{n - 1} , p _{n - 1} ⟩} = \frac{⟨ p _{n} , x p _{n - 1} ⟩}{⟨ p _{n - 1} , p _{n - 1} ⟩} = \frac{⟨ p _{n} , p _{n} ⟩}{⟨ p _{n - 1} , p _{n - 1} ⟩} .

Using part (b) and the Gamma function recurrence,

β_{n} = n (n + \frac{1}{2}) .

The Rodrigues expansion also shows that the coefficient of

x^{n - 1}

in

p_{n}

is

- n (n + \frac{1}{2}) .

Comparing the coefficients of

x^{n}

in

p_{n + 1} = (x - α_{n}) p_{n} - β_{n} p_{n - 1}

therefore gives

- (n + 1) (n + \frac{3}{2}) = - n (n + \frac{1}{2}) - α_{n},

and hence

α_{n} = 2 n + \frac{3}{2} .

Solved by gpt-5.6-sol high.

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