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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

For <monic orthogonal polynomials>, taking the inner product of the recurrence with $p_{n-1}$ gives
$$
\beta_n
=\frac{\langle xp_n,p_{n-1}\rangle}
{\langle p_{n-1},p_{n-1}\rangle}
=\frac{\langle p_n,xp_{n-1}\rangle}
{\langle p_{n-1},p_{n-1}\rangle}
=\frac{\langle p_n,p_n\rangle}
{\langle p_{n-1},p_{n-1}\rangle}.
$$
Using part (b) and the <Gamma function recurrence>,
$$
\boxed{\beta_n=n\left(n+\frac12\right)}.
$$

The Rodrigues expansion also shows that the coefficient of $x^{n-1}$ in $p_n$ is
$$
-n\left(n+\frac12\right).
$$
Comparing the coefficients of $x^n$ in
$$
p_{n+1}=(x-\alpha_n)p_n-\beta_np_{n-1}
$$
therefore gives
$$
-(n+1)\left(n+\frac32\right)
=-n\left(n+\frac12\right)-\alpha_n,
$$
and hence
$$
\boxed{\alpha_n=2n+\frac32}.
$$

Solution

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

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

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

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