 Reversal identity for a finite continued fraction

ID: reversal-identity-for-a-finite-continued-fraction

Because each matrix

M (a) = (a 1 10)

is symmetric, reversing the partial quotients transposes their product. If the convergents of

[a_{0}, \dots, a_{n}]

are

p_{j} / q_{j}

, then the last two convergents of

[a_{n}, \dots, a_{0}]

are

\frac{p _{n}}{p _{n - 1}} and \frac{q _{n}}{q _{n - 1}} .

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