Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / b

Table of contents
- Solution b

Solution

Let

e_{m}

denote the

m

th elementary symmetric polynomial of

t_{i + 1}, \dots, t_{i + k - 1}

. Expanding both sides of the Marsden identity in powers of

x

and equating the coefficient of

x^{k - 1 - m}

gives

a_{m, i} = \frac{e _{m} ( t _{i + 1} , \dots , t _{i + k - 1} )}{( m k - 1 )}, 0 \leq m \leq k - 1.

Thus

a_{0, i} = 1

, while

a_{1, i}

and

a_{2, i}

are respectively the means of the interior knots and of their pairwise products.

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