Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 14A / a

Solution

Let

Q_{k} = P^{(k)}

k

times and using the Leibniz rule gives

(1 - x^{2}) Q_{k}^{''} - 2 (k + 1) x Q_{k}^{'} + [λ - k (k + 1)] Q_{k} = 0.

The induction step follows by differentiating this equation once: the derivative of the coefficient

- 2 (k + 1) x

contributes the additional term

- 2 (k + 1) Q_{k}^{'}

. Thus

λ_{k} = λ - k (k + 1) .

For fixed

λ

, this is negative for every sufficiently large

k

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